Why does shaking my iPhone make touchscreen responsive?

This is not a software issue but a hardware one. You will need to determine if the problem is related to the screen or the logic board. Because this is a used device, you don’t really know what’s it been through.

The first thing to try re-seating the screen connectors. I doubt this is the issue but you have to start somewhere and this one is free :>). The next step is to try a good quality (preferably known-good) replacement screen. You can check out this guide for instructions on how to replace one. It also includes a link to the iFixit replacement screen. If that doesn’t solve your problem, then you are probably looking at a logic board issue. This could be damaged connectors or simply a damaged logic board. This is potentially repairable but will require the expertise of a micro-soldering tech.