iPhone 4 turns itself off when running applications

I got water damage to my phone about two weeks ago. I put it in rice for a couple of days and it worked. There was no noticeable damage to the screen, applications opened, I could send messages and make phone calls, but only when the phone was plugged into a wall charger. It wouldn’t even turn on when connected to my computer.
Naturally, I thought it was a battery issue, so I bought a new one. Now, it works without being plugged in, but once it tries to run applications like Twitter or take a photo or send an email, or even make a phone call, it shuts off. Usually it turns itself back on with a little work, but sometimes it needs to be plugged into the charger to help it turn back on. When it turns back on, it has the same battery charge as before, so the battery is probably good. Any ideas? Does something else need to be replaced?

I do not have AppleCare, though I know they will not cover water damage. The water damage indicators have also turned pink.