It sounds like you were probably given the right diagnosis and it has worsened over time. I'm not sure about natural control of burping but I would recommend basic lifestyle changes to minimize reflux. These include not eating within two hours of going to bed, elevating the head of the bed, weight loss if you are overweight, etc..
My only reflux was burping. Only after being onProtonix for so long and then trying to go off it that I now have heartburn. I've been told that being on the med for long periods of time is not good. I'd like to try to control the burping naturally.
Why would you think that the GERD would just go away? This is equivalent to saying that you just want to stop an anihypertensive medication but can't understand why your BP keeps rising. It sounds like you really do have significant reflux. This will not just go away. As long as your symptoms are well controlled on medication that would be the way to go. If this stops being effective it would be time to consider an anti-reflux procedure such as a laparoscopic Nissen fundoplication. We can generally get 80% of patients off their acid reducers post-op.