From what you described a decent shop should be able to identify the faulty part/structure. If this is something that happened abruptly after never having problems, I would think something broke, slipped or moved enough to make detection not to difficult. If it were me, I'd try to find a shop that actually knows how to investigate and repair. Throwing one alignment after another isn't helping anything. If it won't hold an alignment a quality shop should be able to identify the problem.
A shop that simply says Mustang suspension is crappy and needs to be completely replaced just wants to sell you parts/labor. I'm sure it drove fine when it was new. Something is broken or worn out and it sounds dangerous. Find a shop that can identify/fix the problem.
Anyway, right or wrong, that's my 2¢, good luck and please post back and let us know how it turns out.
I completely agree, I've yet to find shop I trust, especially dealers. I will keep looking to see if I can find a reputable shop around me. I see you have two 2014 GTs in the family. You've never experienced anything like this with either of them?