OK, so if the problem is not swallowing air, what is it? The CPAP pressure is great enough to break the seal created by both esophogeal sphincters and push pressurized air into the stomach (and beyond ). You may have a damaged esophogeal sphincter cause by acid reflux or other injury. The surgery to repair a sphincter is major, expensive, and has permanent undesirable side effects. Not many choose to have it.
So what to try? Quoting from another post of mine,
So for me, tummy sleeping requires a much lower pressure and also solves the aerophagia problem. I can sleep on my sides at the same lower pressure but the aerophagia remains unabated. I suspect that the pressure on my chest and abdomen, creating by tummy sleeping, helps seal the esophogeal sphincters.3. Aerophagia or gastric insufflation (CPAP air leaking into digestive system). You may have some irritation of one or both of the esophogeal sphincters from the acid reflux. Get the acid reflux under control and the sphincters may heal and be able to hold the CPAP pressure back. This could happen, but in my experience it is unlikely, so you are still left with aerophagia. This can be a difficult problem. The solution for me is to sleep on my tummy in the Falcon position, http://www.uarsrelief.com/sleeppositions.html . This has eliminated gastric insufflation for me and it is also a good position for health of the spine.
4. Positional Sleep Apnea (PSA). This means your apnea is most severe when sleeping on your back and may require much higher CPAP pressures. I was titrated by a certified sleep lab at a very high pressure on my back. At home I have software with the CPAP and over several nights was able to titrate to a much lower pressure when sleeping on my sides or tummy. A lower pressure solves the aerophagia problem for some people. What position(s) do you sleep in?
So in summary,
- Get acid reflux (if any) under control,
- Start sleeping on your tummy and
- Evaluate yourself for PSA and retitrate if possible.

BTW, It is a myth that sleeping on your tummy is bad for your back - it is the best way to sleep for health of the spine and back muscles.