If the car shifts into gear but does not move, that sounds like a clutch issue. Most likely the clutch is worn beyond the friction point, that is why it will not move. Removing the trans to inspect the clutch will be necessary if fluid level is good and slave cylinder operates.
If you have any more questions, please feel to ask. Thanks Jay!
Then I would not think you need a transmission. I know the transmission has to be removed to replace the clutch. So, I would not think you need a trans. I have seen clutches fail and the car not move, like you are describing. Sounds to me, if the trans shifts find, but can move. It's a clutch. The clutch disc, pressure plate, throw out bearing and resurfacing the flywheel, should correct your problem.
If you have any more questions, please feel free to ask. Thanks Jay!