Chosen Solution
I bought a 5s off of eBay for $20, not turning on or booting. I replaced that battery and dock connector, and it worked great. The touchscreen was fine and everything worked perfect. When I put it on charge though, the touchscreen is glitchy, touching apps will bring down the search bar and the scroll on safari is all messed up. When I take it off charge it works great tho. Any ideas?
It seems to be the new charging port. When you charge the phone, thats when it activates the charging port the most. I would try reconnecting the charging port and battery connector and turn it on again to test the charger. If thats not the case than it must be a screen issue that was caused by the previous owner.