Have you inspected the GPSR very carefully to see if there is perhaps a piece of crud (small bit of leaf, etc.) jammed in between the edge of the touchscreen and the bezel (case) that surrounds the touchscreen? That is my guess about what could be causing your problem.
Perhaps try this: Fill the kitchen sink up with warm water (body temperature, not hot), add a little dish detergent, submerge the entire GPSR, wash the screen with your fingertips, and if you have a very soft toothbrush around (like what a child would use), gently brush around the edge of the screen, where it meets the case, to dislodge any dirt particles that might be trapped in there.
Then, remove the GPSR from the sink, rinse it under cool running water, and set it on a towel to dry. If your GPSR has a door on the side of it where you insert a CF card (I think the one you own does), open the door after you have patted the GPSR dry, and leave the door open overnight - this will avoid the problem of any tiny bit of moisture causing fogging of the screen. Once the outside of the GPSR has dried fully (after being left alone overnight, perhaps), turn it on and see if it works satisfactorily.
The above sequence of actions has the greatest possibility of dislodging any dirt or foreign matter that might be wedged into a corner of the screen. If the GPSR still doesn't work after that, I think you will have to send it back for repair.
Michael
PS: The fact that the remote does not work suggests to me that the GPSR is already thinking that someone/something is touching the screen - hence the reason why I suggest you wash the thing so thoroughly.
PPS: You don't have to worry about the rubber caps on the USB or antenna plugs being plugged on tightly, because water cannot get in through those openings even if the rubber caps are missing. But, before you wash your GPSR, open the small door for the CF card, and visually inspect the seal for that door, to be sure there is nothing (like dirt or crud) that would prevent the CF card door from sealing up perfectly. If water gets in that door when you are washing it, you will have problems.