Coughing Cat

I adopted a cat from a local rescue organization a couple of weeks ago. She had been living with a foster (or more than one, not sure) and also in a pet display in a store for about 6 months before I adopted her. Basically since she was picked up on the street (according to notes I got from the rescue with her) she has been coughing. She stayed at the vet for a couple days for coughing and not eating and then seemed to recover. But I guess they never figured out what was causing the coughing, because she's still doing it.

I took her in for a check-up two weeks ago and showed the vet a video of her coughing. He said her lungs sounded pretty good, and suggested deworming and xrays. We tried the deworming, but that didn't work. We're going back to the vet tomorrow (presumably xrays are going to be the next step) and the rescue is being awesome and paying for them.

This is two clips of her coughing:

It looks similar to some other clips I've seen of cats with asthma, but it really doesn't sound to me like she's having trouble breathing. More like trying to move phlegm. The first couple coughs are wetter sounding, with the last few being dryer. She seems to be happily eating, playing and sleeping...it's just this nagging cough. I would say I hear at least 2 5-10 cough fits a day (and I'm at work 9-5 so presumably more are happening). I'm hoping she hasn't had a respiratory infection for 6 months!

We also tried switching litter from pine pellets to Swheat scoop with no changes, though they both seem pretty dust free anyways. Currently eating Wellness wet food Chicken, Chicken/Beef, and Turkey. I also can't really time it to a time of day or event. Sometimes it's after playing, sometimes after waking up, sometimes just laying around...

Anyways, just thought I'd see if anyone had any suggestions, and keep this updated with what we find out so we can help anyone else!