My husband also has Barrett's from a lifetime of reflux. Burping is just a part of the reflux problem, and by itself doesn't make Barrett's worse. As long as you have your acid reducers on board, even if you reflux, there isn't enough acid to keep on burning up your esophogus. Since you're still having symptoms, you might want to ask your doc to try a different med. There are several to choose from.
Also, don't forget the lifestyle changes like not eating large meals, not eating 3 hours before bed, sleeping with an extra pillow to keep you propped up, avoiding acid-producing foods and caffeine, etc. Just taking a pill doesn't mean you still get to do the things that got the reflux going, or aggravated it, in the first place. :-)
My husband also has Barrett's, he takes nexium. He also has to have endoscopy's. He burps sometiemes but not alot I dont think. But dont drink water two hrs before bedtime. Major heart burn. Doc also said that.
Thanks for you thoughtful response. I've found the only food I can tolerate is RAW. My son is happy because mom is becoming a vegan. I just had a huge bowl of raw greens with sliced cucumber--not enough to distend my tummy. Ah, no pain. I'm working on a do-able diet, as well as life-style changes. I've noticed that my tummy doesn't hurt when I go for long walks. It relieves the pressure under my sternum. Go figure.
Good luck.