I am not sure about replacing the display or the logic board, that's like using the big guns right away. If you are handy, try to replace the coil on the logic board first. A lot cheaper...you can get the 6R8 coil on eBay and many other places or from here and if you are wondering how to do it, you can watch a video.
Have you thoroughly dried out your iPhone? If not, then you may want to use the Thirsty Bag to get all the moisture out of your iPhone.
If you still have the same problem after drying, then you probably need to replace the display or the logic board.
LCD panel / Does not include touch portion of display or the metal backing plate. — 16 available at 34.95 each.
Drying bag for liquid damaged electronics. The Thirsty Bag is a liquid absorption tool designed to dry all the water out of your phone after it gets wet, protecting it from liquid damage. The Thirsty Bag needs to be used immediately after an accident. Buy — 50+ available at 9.95 each.