A bola-launcher, modified, should do the trick.
Since you're looking for the "whip-slap" effect, you need a projectile that is long, thin, flexible, and hits sideways. Obviously a whip is the first connection, but that's still at close range. The possibility I thought of for a longer range weapon, that has the same kind of strike, is the bola.
A bola consists of rope with heavy weight on the ends. It is swung and thrown, and it hits the target with the rope, letting momentum wrap the weights around the target to tangle. Obviously, since this is a weapon, the momentum behinds the weights striking can do a good deal of damage - not ideal for our purposes, and the reason the concept needs some tweaking.
So, the bola needs weights on the ends to make it easier to launch, otherwise there's no way for a light rope to get enough momentum. A towel would probably weigh less, generally, than a bola - but this is indeed counteracted by the weight of the added water. A towel would also have more mass in the middle, less on the ends. This would not be great for the spin-throw that a freehand bola uses, since the weight differential helps build up momentum, but since we are using a bola launcher anyway, that is less important. I think we will still need weights on the ends, for launching and stabilizing the throw, but they can be smaller.
With more weight in the middle of the towel and less on the ends, and greater air resistance to the volume and mass of a towel compared to rope, this "towela" is probably nudges itself out of the dangerous-weapon range and back into the merely painful category. I'm sort of picturing a longish, narrow towel with each weight connected to both corners on the narrow ends, for the little parachuting imagery it gives me.
On to the bola launcher. A bola launcher, from what I see, uses two parallel (and identical) mechanisms that grip the weights and launch simultaneously when the trigger is pulled. The mechanism could be similar to a (double launching) crossbow, or as complex as any mechanical launcher or firearm.
Our "towela" launcher would need to be somewhat stronger, as a wet towel will be heavy and the distribution of mass would mean more resistance - so more power needed to effectively launch the towelas. The loading mechanism would also need to be modified, towels being thicker and the connections to the weights would be wide and clumsy compared to rope connectors, making it harder to reliably grip and release the weights unless some thought went into it. It would take some practice to learn to balance factors like distance, strength, and materials to make precise effect desired, to quickly load, arm, and otherwise manage the towela ammunition, and to reliably aim at and strike the targets.
Of course, since we're modifying the towelas anyway, there are some extra possibilities. A cotton towel (think tea towels or dish towels) is thinner and lighter and so will need stronger weights than a heavier terrycloth towel, but can generally act more like a bola in strength and force. If the towel is straight between the weights, it will have more air resistance, less force, and more surface area hit than if the towel is twisted (it will then act more like a rope than a parachute). The towela can be loosely or tightly twisted for further control over the force vs resistance balance. And, it may be possible to add some padding to the weights themselves, to minimize the distraction from the blow when the weights land, in order to maximize the snap-whip effect from the towel's sideways blow.