This was one of the hardest parts for Kevin, and me. I kept thinking he was going to choke to death!! The gagging was awful. Lot's of water is what he did. I would check with your doc, like you said, before adding anything OTC. It may somehow make it worse because it's not like phlegm from a cold that you can dry up. At least I don't think it is. Kevin did find that walking around helped him to cough more of it up. Sleeping upright helped too.
Our friend Cathy on these forums and FB gave us some tablets that help with dry mouth. I will look at the name of them again. You're not ready for them yet, but soon will be. You can get them on Amazon. Will let you know.
Hang in there. Try the outside walk. Just a new scenery.
Kathy