Well my initial thoughts of what it could be don't seem right now. Maybe you do just have a dry throat. Some autoimmune conditions can cause this. Some medicines can cause this. Or perhaps it comes from being nervous when reading aloud. Because you said once you do it for awhile sometimes you do not have to swallow. Maybe because you become more confident/less anxious.
I don't think others really coordinate swallowing when reading. But maybe some have to. I know when I was younger I HATED READING OUT LOUD! I would be so panicky! But I don't remember it making me swallow. I just had a hard time breathing.
Maybe you can tell you teacher(s) and see if they make it easier for you. Or ask your doctor. Sorry I couldn't be of more help

. I feel for you...must be frustrating! I hope others might have some ideas.
Bella