Gabapentin and similar drugs tend to have some commonalities re. side effects. The two you mentioned - weight gain and dry mouth - are not unusual. However, if one of these drugs works for you, it's worth trying to resolve any side effects that don't go away on their own.
My wife has to deal with chronic dry mouth and eyes. For her dry mouth, she uses a mouth wash made for dry mouth a few times each day, plus a dry mouth toothpaste. Both are sold under the label Biotene and don't require a script. (There may be similar, but different brand, products available that a pharmacist could recommend.) She also helps her body stay hydrated by drinking at least the minimum recommended number of glasses of water daily.
The weight gain issue is more difficult IMHO. When I was trying different ssri anti depressants to help control my anxiety disorder, I gained over 50 lbs in a year, then finally leveled off. I tried eating slightly fewer calories but that didn't help. Ultimately I switched solely to benzodiazepines for my anxiety and both problems got much better. If I had to do it over again, i.e., take a drug that causes weight gain even with some reduction in calorie intake (but no change in physical activity), I would see a professional dietician now rather than just radically cut calories by 25% or more. My reason for recommending a dietician is that since you're limited in what you can do physically, yet would like to lose pounds and keep them off, I now believe a professional is a good place to start. I suspect, but don't know for sure, that a dietician could suggest a reasonable, lifetime change in your eating habits that would slowly take lbs off and help in maintaining that loss. I know from observation that diets that take off pounds quickly are likely to be ones that you'd find difficult to live with on a long term basis. I base the previous comment on having seen friends and acquaintances go on quick loss diets, reach their goals, then over several months to a year, regain the weight. Good luck.