American and United have both been great. I've flown across the world with a dog on both carriers. No hassle at all: at airports, clubs, etc.
I ensure that all of my paperwork is honest, accurate, and readily available.
American's "SAC" desk has always been helpful. If you have questions, give them a call. They've always been so supportive and thoughtful.
The only snag now is that United banned ESANs from flights longer than eight hours. (I'm not sure about the legality of that, but it just means that I'll fly on another airline.)
I've also flown on Southwest, where they were a bit more fussy. They seemed to do a bit more eye rolling, but they also explained that they've had a deluge fo fake paperwork. On the other hand, the cabin crew on Southwest individually reminded me, "If the masks drop, put yours on first, then put one on the dog." I'm glad they have the policy to remind everyone.
My advice:
- I keep a bound binder of all of the paperwork: vaccinations, letters, and individual forms for each of the US carriers.
- I carry forms for American, Delta, Southwest, and United because I've had cancellations, re-routes, delays. It's much easier to have the paperwork ready in case of a change in airline.
- I'm fortunate to usually fly in first class. My dog is small, but economy class is tight for a flight longer than--say--90 minutes. To be fair, it's tough without a dog.
A couple of weeks ago, I flew for the first time without having to medicated. It was the first time in (I think) 11 years. That's all thanks to the dog. Arf!