Should I redo my bathroom shower or try to waterproof it?
Since the plaster wall (hopefully drywall) has gone soft, I'd open it up. Cut out a section from the floor 2' up (I pick 2' since you can see a lot but still use a half sheet of drywall to repair it, leaving the other half to repair your repair). You don't want to leave drywall that's gotten that wet installed anyway.

For it to get this wet this fast, I can come up with the following guesses:


You've been using your shower as a bathtub, blocking the drain, and getting significant water pressure against the tiles.
The floor is improperly slopped and water is staying against the wall instead of draining.
You have a leaking pipe before the valve. The wall would remain wet or get worse when the shower is unused.
You have a leaking pipe after the valve. If you replace the shower head with a water tight cap and open the valve, you'll notice water without the shower tiles getting wet.
You have a bad seal around some opening in the tile (e.g. the plate that covers the valve) and water is spraying against it. These should be sealed around the top and sides, but open on the bottom to drain out any water that does happen to get behind them.


That said, without a water barrier behind the tiles in a shower, you've got a time bomb. Even if you fix the current problem, the tiles and grout are porous and will slowly leak water into the structure of your home, causing it to rot and mold. They shouldn't leak so fast that you'd see the current problem, but they will leak. A surface water proofing will help, but it won't prevent it.