I'm not so sure. Sheldon Brown argues that riding with drop bars puts your back in an arch, which absorbs shock better than when it's upright. Sheldon isn't a doctor, but that makes sense. In any case, I don't think it's safe to assume that an upright position is better for the back.

My advice would be to try something different from what you have now, and see if it makes things better or worse.

I recently hurt my back, and the key to riding pain free is to listen to my body and pace myself accordingly. If I push too hard it when it hurts, I will hurt all evening.

But most importantly, see a doctor if you haven't already. Back pain could "just" be painful, or it could be a sign of something really bad. Even if it's "just" painful, the doctor can give you meds that work better better than Advil. Since your pain is 20 years old, ask if PT would be helpful.