I have poor balance as well from my meds when they kick in, but I walked before I took my meds yesterday and I was fine. I'm not sure if that is the case for you, but that's just my specific situation. Is the balance thing all the time or only after you take your meds?

You could try an indoor stationary bike. I'm blanking out on the exact name because there are multiple types of stationary bikes, but I mean the ones where you sit-up perfectly straight rather than lean back.

You could also do light weight training. I promise you won't become a hulking beast. See this fit girl who is thin, but EXTREMELY strong from weight training:

No bulging muscles. Just lean and fit. She does calisthenics and some weight training. Lifting weights should be ok for you, I think. I would just start off with 5 lbs weights, do some pushups, do some crunches, etc.. That'll burn fat because muscle needs the calories from fat and your muscles will start chewing up the fat. You'll probably get sweaty and your heart rate will go up as if you were doing cardio if you do the weight training without many breaks.