I had a horribly dry mouth for the first month I started HRT, along with a really unpleasant taste. It would also wake me up, so I kept some 'artificial saliva' by the bed. Xylitol is supposed to help, but unfortunately it upset my stomach - there are lozenges that you can keep in your mouth overnight, but I can't remember the name, sorry - xyli something? I'm prone to saliva stones, so I now regularly massage my saliva glands to increase saliva flow (found a youtube video) and luckily it has settled down, and although it still flares up every now and then, it's nothing like as distressing as that first month. Maybe as your hormones settle it will also ease off for you, although I didn't have the problem you describe with swallowing. That sounds very unpleasant.
Anxiety is really distressing. Following recommendations on MM I've recently been using the youtube channel 'therapy in a nutshell' which has some excellent grounding exercises for anxiety. There's also another thread about 'the tapping solution' app - you could maybe try the free trial and see if that helps you at all.
I'm on estradot patches, so can't comment on elleste. Sorry that you're suffering.