I had a similar problem. My iPhone SE got wet (not submerged) in my backpack biking home in the pouring rain. The screen was visibly damp near the bottom so presumably the water got in through the speakers or ports. Despite some speculation on the interweb, the SE does not appear to be water resistant to any great extent, so don't be dumb like me and keep your phone dry! I briefly tried the standard advice of both putting it in a container of rice and a baggie with random dessicant packs. I've got a small humidity gauge which indicated that neither of these environments was any drier than the ambient air in our house, so I settled for just leaving it on the counter. It got pretty warm (which was probably the battery shorting out), the screen went all screwy (double images, etc), and the home button was stuck on. After about 2 days the screen was back to normal and the phone would turn on briefly but would not charge, and the home button didn't work at all. Since both the screen and main board seemed to be...
I agree with the last point about the need to get the top of the screen to seat flush before pressing in the rest of the screen. There are three plastic tabs on the top of the screen which need to fit into three holes in the metal case. My plastic tabs were bent up (upon opening?) and I had to bend them down (using the blue ifixit opening tool) to get them to fit into their holes so I could seat the top of the screen.