The drive chain comes off while going forward and it does not seem to have much slack, the gears look good, and I can't find any adjustment.

Hi,
My guess is that the chain is compromised in some way at one point and on one side somewhere in its circumference. I would replace it and see what that does. A chain that has a weak spot on one side will flex at that point in directions not allowed for proper operation. The parts breakdown is here:
Murray st Parts Diagram Index for G2784-000
Peace,
Clarke