Hi. I have a lawn boy with a simmilar issue. But, the drive won't work unless you pull it backward to re-engage the one wheel driving.
I take it you're one of the people who leaves your self-propelled engaged most of the the time? I let up on it when I am going to back up, or turn sharply. I don't just cram the drive engage on, and hold it till I'm done mowing....not saying you do...but if you are one of these types, that puts a lot of stress on that flimsy differential - and that seems to be the problem, based on your description.........
Even if you don't - sounds like the diff has pooped on you........(or is in the process of pooping).
Check ebay or craigslist for a good diff - otherwise, a new unit may carry a hefty pricetag. Or, run it till it won't move by itself anymore - then worry about a replacement. Up to you. I'd be shopping ebay, myself, for a new dif----