How would you know if transmission is bad.

The clutch totally went out, I could try to pump the clutch pedal a few times and it will shift but it is still difficult to shift sometimes, I believe that pressure plate is bad but not sure, anyways when they checkedthe transmission fluid it was dark and hot and got some junks in it. They said they came from the synchronizer and needs to be replaced. It would cost about 1200 to rebuild the transmission. I couldn't afford it so I drove it back. I could still drive it. But the only thing was that, the shifting was very difficult since the clutch was bad. If I replaced the clutch, will the problem go away? And if I don't replace the synchronizer like they said, will it ruin the new clutch?