My DD throws this pitch, and throws it pretty well. The "arm/hand shaking" is a defense mechanism that I have seen a bunch out of girls this age. Throw a bad pitch, and they shake their hand. So, I am guessing that is the response that you are seeing.
I have seen this a lot myself. My DD use to do this after a bad pitch. I think sometimes they do it and don't even realize it. It was a habit DD had to break. I basically told her when she does this she is telling the other team she threw a bad pitch and what she needed to convey was that she meant to throw the pitch in that spot, even if she didn't.
As far as wear and tear with a drop ball...DD throws it a lot and I have not seen any issues over the past few year. I think you will be surprised that even at 14U your DD can be successful with either a FB or Drop and a good CU. I would keep working hard on the CU. I think it is a must at 14u. If she can learn to place that drop inside and outside and a different levels she can be very successful with just that and her CU. Once she learns to control her drop some throw it with a "loose wrist" and she can get movement like a DC. As she learns the Drop this will happen some without even trying.
Again, drop is a great pitch...but got to have a CU at 14U IMO.