Yes.
The actual rules for this are oddly worded, but the notation is pretty simple.
In general, there are two formats for "things that increase your AC." The most common format is the one attached to armor:
[your AC equals] 11 + Dex modifier
or Draconic Resilience:
your AC equals 13 + your Dexterity modifier.
Note that there is no "plus" at the beginning of the formula. These items set your AC to a specific value.
The other format is that used by shields and class features that increase your AC above its base value:
Defensive fighting style:
You gain a +1 bonus to AC.
Or a shield:
Wielding a shield increases your AC by 2.
These are phrased in terms of increasing, rather than setting your AC. They are preceded by the phrase "increases" or a plus sign.
So, it's pretty simple. You get one thing that sets your AC, and any number of things that modify it (except where restricted by other rules, like the rules stating you can only use one shield).
A fighter could very easily have an AC like this:
[15 + Dex Mod] (half-plate) [+2] (shield) [+1] (defensive fighting style)
For a total AC of 18 + Dex Mod.
But he couldn't add a chain shirt to increase his AC, because both that and the half-plate set his AC to a fixed value.
The same applies to Draconic Resilience. You can stack it with a shield, because a shield increases AC. And you can further stack on the Shield spell, giving you a (temporary) total of [13 + Dex Mod] [+2] [+5], or 20 + Dex Mod.
But you can't combine Draconic Resilience with Mage Armor, because both Mage Armor and Draconic Resilience are trying to set your AC to a fixed value.
Some things Draconic Resilience stacks with:
Some things it does not:
RAW
The rules for AC are on page 14 of the Player's Handbook, or page 9 of the Basic Rules.
The relevant part is this:
Without armor or a shield, your character’s AC equals 10 + his or her Dexterity modifier. If your character wears armor, carries a shield, or both, calculate your AC using the rules in chapter 5. Record your AC on your character sheet.
[...]
Some spells and class features give you a different way to calculate your AC. If you have multiple features that give you different ways to calculate your AC, you choose which one to use.
The rules called out above appear to be these:
Wielding a shield increases your Armor Class by 2. You can benefit from only one shield at a time.
If you wear light armor, you add your Dexterity modifier to the base number from your armor type to determine your Armor Class.
(and so on, for each type of armor)
The rules are sloppy here. Technically, they don't give you a calculation for AC when wielding a shield and no armor. And they never explicitly call out what is or is not a "way to calculate your AC."
With that said, we can use examples to build the intent of the designers.
It's pretty clear that you don't have to choose between a shield and plate. It's also pretty clear that a shield shouldn't set your AC to 2.
Finally, it's pretty clear that wearing a chain shirt with half plate shouldn't set your AC to 28 + 2 * Dex modifier (max 2).
It seems obvious then, that a "calculation" is something that sets your AC to a value plus other modifiers.
If something just adds on to (or increases) your AC, you continue to use whichever calculation is most favorable plus the new modifier.
Sorcerer has very little to offer an Arcane Archer-type build, with the exception of the Elemental Affinity feature from the Draconic Bloodline origin. Combined with Elemental Weapon, this allows you to add your Cha modifier as elemental damage to every shot you fire from your crossbow. The problem here is that Elemental Weapon is restricted to the Paladin class only.
So you have 3 options, explained below. Regardless of which you take, you'll be focussing on Cha and Dex, with Con as a secondary as usual. So you'll want to spend your 27 points in the 15,15,15,8,8,8 pattern. Then Half-Elf brings you up to 17 Cha, 16 Dex, 16 Con very neatly. Each build gets you 5 ability score increases over your career. You'll want to use these to bring Cha and Dex up to 20 and take Crossbow Expert so that you can dual-wield hand crossbows.
Option 1 is to take at least 9 levels of Paladin so you can cast Elemental Weapon. Then you'll want to bring Sorcerer up to 8 and Paladin up to 12 so you don't lose ability score increases. This isn't a terrible idea, as many of the Paladin's [X]ing Smite spells work with ranged attacks, making a pretty reasonable Arcane Archer. With the Oath of Devotion's Sacred Weapon ability, you could also add your Cha modifier to your attack rolls. That said, many of the Paladin's class features don't work with ranged attacks, and it's not exactly an Arcane Archer anyway. Also, you'll have to have Str 13 to multiclass Paladin; so your starting ability scores will be 15,15,13,12,8,8 with Half-Elf making it 17 Cha, 16 Dex, 13 Str, 13 Con. (You only lose out on Con, so it's not too bad.) The main thing this option gets you that the other 2 don't is the cool Smite spells that add damage and additional effects to your crossbow attacks, just like an Arcane Archer.
Option 2 is to take 6 levels of Lore Bard for Additional Magical Secrets, which you can use to gain access to Elemental Weapon. You could use the other magical secret to get Branding Smite for that Arcane Archer flavour. Then, you'll want to take 5 levels in a martial class for Extra Attack; I'd strongly recommend Fighter, so that you can take Fighter 6 and not lose an ability score increase. Once you're Sorcerer 6/Bard 6/Fighter 6, take 2 more levels in whichever of these 3 classes you like to get your last ability score increase. This option is the least focussed, but you get the Archery fighting style and Improved Critical for your crossbow, so it's got a slight edge with the crossbow you're looking to optimize. This can also get you Arcane Archer-ing the fastest, since your combo comes into play at level 12.
The final option is to take 10 levels of Valor Bard, and use Magical Secrets to get Elemental Weapon. The other magical secret could be used for Banishing Smite, Branding Smite, or Staggering Smite for more Arcane Archer flavour. While this doesn't sound as good as Lore Bard, you get Extra Attack for free along the way. Then you should probably take Sorcerer up to 8 and Bard up to 12 to keep all your ability score increases. This option gets you the most powerful spellcasting of all of them, but you don't get the Archery fighting style.
Best Answer
You have several options to increase your AC.
Spells
Shield will increase your AC by 5 for a round and it's a life saver if you suddenly get swarmed. Though if you are not careful it can quickly eat through your few spell slots.
Mage armor will set your AC to 13+Dex if you are not wearing armor. It is of no use to a Draconic Bloodline Sorcerer, but it's worth listing for the sake of other sorcerers.
Honorable to mention blur and mirror image. Although neither increases your AC they both make you much harder to hit. Mirror image doesn't even require concentration.
Racial Traits
Beasthide Shifters from ERLW gain a +1 bonus to AC when they use their Shifting trait.
Githyanki from MTF have light and medium armor (but not shield) proficiency through their Martial Prodigy trait.
Githzerai from MTF can cast shield once per long rest starting at level 3 through their Githzerai Psionics trait.
Hobgoblin from VGM have light armor proficiency through their Martial Training trait. It is of little use to a Draconic Sorcerer, but it's worth listing for the sake of other sorcers.
Lizardfolk from VGM have AC equal to 13+Dex through thier Natural Armor trait. It is of no use to a Draconic Bloodline Sorcerer, but it's worth listing for the sake of other sorcerers.
Mark of Sentinel Humans from ERLW can cast shield once per long rest through their Guardian's Shield trait and add shield of faith to their spell list through their Spells of the Mark trait.
Mountain Dwarfs from PHB have light and medium armor (but not shield) proficiency through their Dwarven Armor Training trait.
Tortles from TTP have AC equal to 17 through their Natural Armor trait. That just speaks for itself.
Loxodons from GGR have AC equal to 12+Con through their Natural Armor trait. At first glance this seems less than 13+Dex, but being able to scale both your AC and HP simultaneously with Con, while disregarding Dex can make you quite tanky.
Simic Hybrids from GGR can gain a +1 bonus to AC while not wearing heavy armor at level 5 through their Natural Enhancement trait.
Warforged from ERLW have +1 bonus to AC thanks to their Integrated Protection trait.
For races that appear in multiple books, I only mention their first official publication.
Armor and shield
If you want heavy armor and shield, you will have to multiclass, the typical choices being a dip into either Fighter (which has to be your first level, but also gives you the Defense Fighting Style) or Clerics of certain domains.
For medium armor and shield you have a greater variety of choices.
On one hand, you could multiclass, the typical choices being again a dip into Fighter or Cleric.
On the other hand, you could get the Moderately Armored feat at level 4. However, to take that feat you need to have proficiency in light armor, which you could acquire from your race: Githyanki for their Martial Prodigy trait, Variant Human to start with the Lightly Armored feat, Hobgoblin for their Martial Training trait, and Mountain Dwarf for their Dwarven Armor Training trait.
What to choose
Each of these options come with costs, so which you choose depends on what you want to prioritize:
In your case, because your stats are all very high and you don't have to worry about being MAD, I think that dipping Cleric with a domain that grants heavy armor is the most effective option. You could do just fine with any race, but to maximise the AC further I would pick the Warforged.