There is a difference between a spell that says "You can make melee touch attacks", such as Produce Flame and spells (for example, Shadow Walk) that have as their target "Creature(s) touched".
The rules you quote above, where you can touch up to 6 creatures, refer to the latter; that is, spells that have "creatures touched" in their Target: line
Spells that allow you to make Melee Touch Attacks use the regular rules for making melee attacks, and will only allow attacks based on your BAB. Which is 1 per turn for 6th level Druids.
Flurry of Blows modifies your full-attack
It adds additional attacks, and applies a penalty. It is otherwise a normal full-attack like any other.
This means you may use any weapon you have available for any given attack. If you are holding two weapons, you can use one or the other for each attack you’d normally have.
Note that this is exactly the same as when you don’t use Flurry of Blows. If you have multiple attacks for whatever reason (BAB +6 or higher, the haste spell, whatever), you can use any weapon you have available for any given attack as you please.
On the flip side, you do not have any more attacks, normally, than you would with a single weapon. You simply get the attacks you would otherwise have. Those can be the ordinary attacks of your full-attack, or extra attacks from Flurry of Blows, or whatever. Simply having two weapons does not give you extra attacks.
To get extra attacks, you have to use the Two-Weapon Fighting combat option (and then you are subject to its penalties and restrictions). In particular, Two-Weapon Fighting is an exception to this “any weapon for any attack” bit. Note that this is more than just having two weapons; you have to actually choose to use Two-Weapon Fighting, or none of the following applies. When you do choose to use Two-Weapon Fighting, it changes your full-attack (much as Flurry of Blows does): it grants an extra attack and applies a penalty, and it also applies a special restriction on which weapon you can use for which attack: specifically, your bonus attack from Two-Weapon Fighting has to be a different weapon than the one you are otherwise using.
Furthermore, under Pathfinder rules, Flurry of Blows is Two-Weapon Fighting (except it doesn’t actually require you to use two weapons). As such, the idea is you cannot use it and Two-Weapon Fighting at the same time. If this is the case, there is absolutely zero benefit to having two weapons, except I guess the ability to pick which one you want. You’re better off with a more-powerful single weapon than splitting your wealth between two.
Unfortunately, the rules here are notoriously vague and have required numerous amendments, explanations, Paizo contradicting themselves several times, and finally overruling previous rules entirely because they never made sense. This is actually one of the most famous fiascoes in Pathfinder history, and the rules are left a mess as a result. I suggest just returning to the 3.5 rules, before all this started: Flurry of Blows and Two-Weapon Fighting are completely separate things, combine them if you want (taking even more penalties and gaining even more attacks). This is still a bad idea, since it is a very weak option, but at least the rules are clear, make sense, and there’s actually, theoretically, a benefit to dual-wielding as a monk.
Best Answer
Oh! Well, that turns it from some complex mass of edge cases into a simple proposition.
If you go by the completely psychotic RAW or the Wizards FAQed answer on natural+flurry, then the flurry would not drain but any additional natural attacks would. Of course you'd have to get additional natural attacks, which just being a human soul eater or whatnot wouldn't get you.
If you go by the more reasonable interpretation accepted as the answer on the other question, and that's how I'd rule, then I'd also rule that sure, it's a flurry but "with" natural weapons and so totally, the energy drain would proc on each. It says "touch" after all, and considering an unarmed attack not a "touch" stretches in-game-world credibility.