Play in gear shift

It could be the clutch or transmission. Having a transmission rebuilt can be more cost effective than a whole new one. You will have to check out what's available to you and consult your budget for the repair.

change filter could help for a minute but clutches are fried in drive

run fault codes and look for a TCM problem (transmission control module). Check fluid level in transmission

make sure you tested the level correctly. Sounds strange to be high, unless it got flooded.

seems like your shift cable is disconnected , you might have hit it when you were working on the alternator just go back and re-trace your steps

transmission shifting solenoid(s)... could be a broken wire or bad solenoids.
best regards
Ck

valve body is sticking in the transmission

it sound like you have a vacum leak or your shifter cable may need adjusted

Depending on whether it is a column shift or floor, you have to get the linkage loose enough to get the car in neutral, move the selector back to the proper setting, and tighten everything up.

If the car is not setting on a hill or incline, you should be able to move the shift levers by hand when you've loosened the linkage and move it until it is in neutral. Put on jack stands and chock the wheels that are on the ground.