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 working, I decided to take a stab at fixing it. Running with the assumption that the battery was toast, I ordered [product|IF327-004-4|ifixit's battery replacement kit including tools] which includes some handy items for getting the phone apart. Most (like the suction cup and prying tools) could be scrounged from home, but the pentalobe screwdriver is essential and the price of the kit is pretty reasonable.
Once I had the phone apart (following ifixit's directions), an inspection of the home button connector revealed some corrosion. I scrubbed both male and female sides with a toothbrush and 91% alcohol. I didn't see evidence of corrosion elsewhere (though I didn't look too deeply), and knowing that the phone only got damp decided to skip/risk not removing and cleaning the main board. I replaced the battery and reassembled the phone, and it appears to be fully restored for now.