How do you tell if it is just your gimbal bearing that is worn out, or if your whole assembly needs to be replaced?

The bearing will give off a grinding/growling noise when bad. You'll know it.
When the cost to rebuild nears the cost of a new unit, buy new. Look at all the parts for wear and corrosion to determine what is usable.

I was going to rebuild both of my Alpha units, new bellows, bearing, shift cable. Once apart, the swivel pin was pitted from rust and the gimble ring had a small amount of wear. I could have repaired both of those items myself, but the time involved and the other parts cost made it not worth rebuilding. Mine are on a cruiser that lives in the water, so they had some amount of corrosion that I just couldn't justify rebuiding them. I found 2 new, never installed complete transom assemblies on ebay for $600 each.
Now EVERYTHING is brand new.

How do you rebuild a gimbal housing, and are the TRS housings rebuildable? I always thought that they were junk after they go bad...

No experience with TRS's, so can't answer that one.

Where is a good place to pick up the "kit" to repair this issue? I have about an inch of play in my drive where the drive hinges on the gimbal on one side.

Unsafe and needs to be fixed.