I've seen this before as well. I'm an iPhone tech for a living and work on a lot of water damaged iPhones. If you have cleaned the motherboard really well and tried a new screen assembly as well....Then the problem is with the motherboard, Most likely the chip that controls the touch activity. Unfortunately, this is a machine soldered part thats going to be near impossible to replace unless you have specialized equipment and experience. So....Most likely not much can be done. I know of a few shops, but non that replace those types of electronic components on motherboards.
I have seen business advertising and selling service on eBay for things like backlight repair, power connector re soldering and such. But I dont know If any are good and reliable or not. I tend to not trust those types of things and stick to fixing it myself.
Hope this helps