Grey wifi, won't charge, tried battery... what else?

Hi! I've got an iphone 4s that had a minor water incident. It wouldn't turn on, I took the back off and dried it as well as I could. It was only wet near the top. I left it for a day in rice (amateur move, I know now.) and then I charged it. It seemed to work fine, except for getting a teeny bit hot. Then about three days later, it broke completely. The wifi button was grey, it was constantly "searching," (I don't have a mobile plan) and it would only light up or do anything at all if the charger is plugged in. When the charger is plugged in, it says 100% battery. But as soon as I unplug it, it gave me a black screen. I bought a new battery, replaced it, and immediately I had a working iphone again!

Everything was back to normal. It was still a little bit hot, just a little, but I thought maybe I had incorrectly "Seated" the battery. (I couldn't figure out the corrrect way that the cable was supposed to sit when I put the battery in.) I used it for 2 days, glad it had been such an easy fix. I charged it once in that time. However, last night I let the battery run down to 1% and it powered down. When I went to charge it this morning, it's doing the same things it did before I changed the battery! Oh noes!

I'm willing to replace some parts, but honestly, I don't sauter... Is that the word? I only know screwdrivers and spludgers. Does anyone have any hope or knowledge of what I need to do to this bad boy?

Thanks so much.