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...
War Magic Beginning at 7th level, when you use your action to cast a cantrip, you can make one weapon attack as a bonus action.
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...
- In Tier 1, GFB adds a small amount of damage to a secondary target. The problem is that GFB uses the Cast A Spell action, which doesn't trigger the bonus action attack from using two weapons (you have to use the Attack action). If you're using two weapons, GFB actually lowers your DPR. Assuming a d6 weapons in each hand with the fighter style and two hits, but no feat, you're looking at damage of 2×(1d6+attack stat) with casting stat in damage to a secondary target.
- In Tier 2, a fighter gains Extra Attack. While GFB does increase to an extra 1d8 to the original target and the secondary, you're also likely to have Dual Wielder by this point and be swinging d8 weapons.
- GFB hits for 2d8+attack stat to the primary and 1d8+casting stat to the secondary.
- TWF hits for 1d8+attack stat to up to three targets, each of which can potentially be a critical hit.
- At L7, an EK eeks out a bit more, getting the hit from GFB, damage to a secondary target, and another basic weapon attack from War Magic. Note, however that War Magic is a swing with the same weapon. You don't even need two weapons for this, nor do you need a feat.
- In higher tiers, the fighter gains additional attacks, but GFB gains two damage dice each level. Unless you're crit fishing (or need high rolls to hit at all), GFB is a good choice. If you're having a hard time hitting, more rolls is better (because any damage is good damage), so you wave off GFB.
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.
Ultimately yes, with some caveats that I think may not be what you are intending.
The Eldritch Smite warlock invocation (XGtE, p. 56) reads:
Once per turn when you hit a creature with your pact weapon, you can expend a warlock spell slot to deal an extra 1d8 force damage to the target, plus another 1d8 per level of the spell slot, and you can knock the target prone if it is Huge or smaller.
This is an extra die or dice that is/are added to your damage roll, the same with Hex:
Until the spell ends, you deal an extra 1d6 necrotic damage... PHB pg. 251
"Damage rolls" are specified on pg. 196 of the PHB:
Each weapon, spell, and harmful monster ability specifies the damage it deals. You roll the damage die or dice, add any modifiers, and apply the damage to your target.
Since each ability specifically states that your "damage roll" and not "damage die/dice" are getting the boost, this indicates the following:
It does not add an extra +2 to each die that is rolled, just to the overall result of all dice rolled for damage. This also applies to Hexblade's Curse:
You gain a bonus to damage rolls against the cursed target. The bonus equals your proficiency bonus. -XGtE pg. 55
They can stack though, with both being applied at the end of all damage rolled.
Dueling in the PHB on pg. 72 reads as such (emphasis mine):
When you are wielding a melee weapon in one hand and no other weapons, you gain a +2 bonus to damage rolls with that weapon.
Also of note is that if you are wielding the Polearm two-handed, it does not add the extra damage, as it is explicitly stated it must be in one hand.
It should also be kept in mind that the Eldritch Smite invocation is boosting the damage of an attack you are already making, so it is not Eldritch Smite that is getting an additional damage die by Hex but the damage roll while using Eldritch Smite.
The end result of all of the above being used at once, assuming you are using a Quarterstaff as your pact weapon since that is one of only two polearms (the other being a spear with the same damage die and different damage type) that can be used one handed, would be on a hit your damage roll consists of the following: 1d6 bludgeoning (Quarterstaff), 1d6 necrotic (Hex), 2d8 force (Eldritch Smite [assuming level 1 slot]), +2 (dueling style), +2 (Hexblade's Curse adding proficiency bonus assuming low level in both classes), +3 (an assumed value for your Strength modifier)
Best Answer
No, you only double damage dice
The rules on what critical hits do are found on page 196 of the PHB (bold added):
Damage done via something other than dice (like adding your Charisma modifier from Hexblade's Curse, or the +1 you would get to damage from the Improved Pact Weapon invocation) is not increased by a critical hit.