can you connect to it with VNC? (perhaps you may need to launch the VNC server via SSH)

If you can, then try installing HealthCheck , then once you have it installed, run it, go to the Tools tab and click on the Accuracy test button, once it's running try draging your finger all over the screen and via VNC see if the crosshair is in the correct position.

I dunno if you can replace the screen without replacing the digitizer at the same time though