Have you ever changed the trans fluid/filter on it? If not you could try that. A trans-go shift kit could alleviate the 1181 max adapt shift issue. You could try disconnecting the electrical plug on top of the transmission, cleaning both sides with electrical cleaner spray and reconnect and see what you get.
The shake at 50-60mph could be a front wheel hub, cv axle, or both. My brothers 01 GT had both axles get so bad it was shaking the dash. I just installed 2 new hubs and cv axles and all is smooth again.
What power steering issues do you have? The only super expensive thing is the rack. All other stuff is cheap and easy.
A trans shop should be able to hook a scanner up to it and command/cycle those shift solenoids I would think. I would call and ask first to be sure that is even possible on an 01.