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1. most likely the seat safety switch has become disconnected or the connections are dirty. The mower is made to shut off if no one is seated on the tractor and it is put into gear or the blades are turned on. Right now I believe that the tractor thinks that you are not seated on it. Check this safety switch and it's wiring.
2. just because the muffler has been welded to the entrance pipe does not mean that the baffles inside the muffler are in tact. I would just replace the muffler with a new one.
For issue #1, I have to be seated on the mower even for it just to start, so even if I am sitting on the mower when I crank it could this still be the issue when trying to put it into gear?