Hi Adz,
I too have an Amiga 2000HD professional computer. I know that the Amiga 2000HD is a much better computer then those TOY Amigas (500, 600, and 1200). Answer to your problem: Did you check the battery area on the motherboard for corrosion or discoloration? If you have not yet removed the battery, you should do so. Also, one of the 8520 ICs may be bad. I can ship you a new pair of these chips for only $20.00 plus shipping. Does this problem occur with all your games? Also, get yourself a WICO Bat-Handle joystick from eBay. WICO made the best joysticks. They usually last for over 5 years of heavy use (mine did).