Engine dies - need help please!

Engine starts fine, with full tank of gas would run forever, but when you engage the transmission or the tiller tines, it immediately dies. If you disengage as the motor starts to die, then it perks right back up and keeps running. When the engine dies, you can restart immediately and it runs fine. If the engine is running and you push the tiller across the yard, it starts to run rough. When you stop pushing the tiller, the engine again runs fine. Neighbor is small engine repairman, and has gone through the carbeurator twice, changed spark plugs, and he thought it might be governor but has not tried that as yet. Has given up in frustration. Does this sound familiar to anyone out there?
I know if I took this problem to a shop, they would try 6 different things and charge for each one - not there yet. Would like to solve this problem myself without doing that.

Engine dies - need help please!