You are making a few mistakes in your build, and in your analysis
You calculated some things wrong, and also your character is not optimally built.
Calculation
You have compared your average damage with the nova damage of the Paladin. A 5th level Paladin cannot do 40-50 damage on each of his turns; only when he decides to spend a spell slot on Divine Smite. This is a finite resource, and he has only 6 of those on level 5.
If he uses Great Weapon Master, it greatly decreases his hit chance, and yet you only compared the rounds when you both hit with every attack. This distorts the calculation.
Build
Fighting Style
Dual Wielding is a bit ahead of any other style before level 5, but quickly falls behind Great Weapons from level 5 in raw damage. As a half-elf you could have put your Ability Score Increase and 15 in Str instead of Dex, making Great Weapons viable for you too.
With the Shield Master feat, even Dueling is ahead of Two-Weapon Fighting in DPR, while greatly improving your survivability, and usefulness for your party.
Feat Selection
Dual Wielder is weaker than an ASI, especially if you are building with Dex as your primary ability score. A +2 Dex to would give you the same average damage increase (1d6+4 = 1d8+3), the same AC increase, but would also increase your to-hit chance. Besides this, it improves both your Stealth and your Initiative.
Keeping up
Multiclassing can help, but not much.
Taking 1 level of Warlock for Hex is problematic with your Cha, but 2 levels of Ranger for Hunter's Mark would work. You could get a lot of mileage out of +1d6 on all your hits, but moving the spell to the next target is a bonus action, just like your off-hand attack. So the faster you kill the enemies, the less you can use the spell. At least with your Con save it is hard to end your concentration.
Taking levels in Rogue is a better idea, especially once you have the Riposte maneuver. Sneak Attack can trigger once per turn, so you can use it with your Riposte.
Note that everything listed here could be done with a Duelist, just better.
Best Answer
I'm not going to do a deep dive into specific mathematics here, because you're already heading down a non-optimized route. What I am going to do is try to give you a feeling for the variables involved and where you should go in the future.
Weapon Choice
Considering the racial choice locks you out of effective use of Heavy weapons, there's no debate that using two weapons is the best choice for weapon DPS. As for which weapons, or if weapons are even the right answer, that really depends on how much more Warlock you're going to take vs. how much Eldritch Knight.
For this character, magic shortswords would be better here, because you could use them with Dexterity via finesse and together via light... but you've got what you've got. The difference between 1d6 vs. 1d8 weapons averages a single point, so it's almost not worth considering if you're not super-optimizing. The difference of +1 to hit is far more significant to your damage output, though - use the +1 longsword for now.
If you're committed to Dual Wielder at some point (perhaps for the aesthetics), twin rapiers is a better option. For survivability though, you may want to stick to one and use a shield. Right now, you could use the sword two-handed via versatile for 1d10, but again that's just one more damage on average. A corpse deals no damage, so unless this character has weapons in both hands, it should have a shield in the other. (Great Swords and Great Axes in conjunction with feats have other benefits, but they're no-go for a Halfling.)
Warlock
Right now, the character can effectively use one of those longswords with Charisma via Hex Warrior. It's not as good as Dexterity, but it is better than Strength. You can't use the other one simultaneously without taking Dual Wielder, because they don't have the light property.
If you go to Warlock 3, you'll be able to pick up Pact of the Blade, which does allow you to use two weapons with Charisma (one via Hex Warrior, one as your Pact Weapon). Of course, you'll need Dual Wielder for that.
Whatever you do, don't neglect Hex. An extra 1d6 of damage on every hit will add up quickly.
Eldritch Knight
If you take Eldritch Knight to 7, you'll gain...
That will let you take the most advantage of green flame blade (herein "GFB"). For the purposes of conversation, I'm going to look at Eldritch Knight exclusively...
Flame Tongue, Frost Brand, etc.
If you're going for big damage as a fighter, you want a weapon that deals extra damage on every hit. Your extra-on-a-crit weapon is nice, but critical hits aren't your bread-and-butter. The dip of warlock helps because of Hex, but full-time-extra-dice is something you really want.
Summary
If you're leaning more heavily towards Eldritch Knight (at least 7): Get a more appropriate weapon, and don't bother using two. Use a shield instead.
If you're leaning more heavily towards Warlock: Get a more appropriate weapon, and go ahead and use two. Consider leaning more towards Eldritch Knight.