After trying this tips shared by Hypex at AmigaWorld: mouse short cuts. LAmiga Alt for left mouse. RAmiga Alt for right mouse. Ctrl + LAmiga + cursor should move. Shift for high speed. IIRC, and seeing that the cursor moves fine, I'm guessing that I may be in need of a new mouse. Unfortunately, I don't have a spare one to try.
It is possible you need a new mouse, or it could be a problem with paula. The keyboard shortcuts don't really tell you anything as that is done in software. It just means the keyboard is working.