Still working on the house. Today I am restaining grout. Oh that is a fun job. NOT.
anyway, Brody did the exact same thing one time after he chewing on a bullystick and he actually had a string of it stuck between his back teeth! It would not come out! It was slimy and of course he wouldn't leave his mouth open for me to get ahold of it. So I got some pliers and hubby held him and I was able to get a grip on the end of the string and pull it out from between his teeth. It was really stuck. So her making those mouth movements makes me think that you should check between her teeth really carefully!!! There might be a little something caught there.
Otherwise I agree with your vet and the chicken neck may have scratched her esophagus going down. Poor baby. I hope she feels better soon. I hope you didn't get a lecture from the vet.
Is she a greedy gus? If so, you might want to feed BIG pieces that she will really have to settle down and gnaw on, like a beef or pork rib with lots of meat? A chicken neck might just be too small for her if she is a gulper.