I have had that happen to me, but I have always blamed it on my puffers for asthma. I figured I must not have rinsed my mouth well enough after using. Another thing to ponder.

I had burning mouth for a year and a half. Eventually realized I had a decayed wisdom tooth. Once it was removed the burning went away. Still not sure what caused what, but I wish someone told me to go to the dentist sooner. I just assumed it was Fibro. That may not answer your question but if it is tooth related it would make sense it could be on just one side. Either way, hope you find some answers! That can be so so painful and scary!