What should I do if being full (eating meals) easily triggers mild asthma attacks?

Possibly... ...It's reflux (heartburn) that is triggering the attacks. Ask your doctor about possibly treating that.
Smaller meals... Strategy would be to eat smaller meals more frequently to eat the same amount of calories, nutrients, etc without getting "full." you should have these sxs evaluated by your doctor to be certain asthma is the etiology of your sxs and not some other entity which would have a different treatment altogether.