Okay Bob. The thing I find a little confusing about your issue is the fact that it starts after you've been playing awhile, but nonetheless, I'll carry on.
I don't have the rubberbanding issue myself (lucky me) but I can make it happen by pressing either of the W, A, S or D keys, the cursor control keys (arrow keys) or the arrowed numberpad keys when scroll lock is off, and moving the mouse at the same time.
So I have written a small AutoHotkey program which forces the WASD keys to type out their respective letters instead of moving the cursor. The program also disables the cursor control keys (arrow keys) and ensures the scroll lock is switched on so that the numberpad arrow keys type their numbers instead of moving the cursor.
Because this program works on my system doesn't mean it will work anywhere else, but there's no harm in trying it.
You can download the program from HERE Bob. Once you have downloaded it, locate it in your downloads folder and double click it to run it. When it is running you should see a green icon with a white 'H' in your system tray. The program doesn't affect how the aforementioned keys perform in any program other than WGT Golf.
When you have tested out WGT to see if it has any effect on your issue, you can stop the program by pressing CTRL + X, or right clicking on the system tray icon and selecting 'Exit' from the menu. The system tray icon should then disappear.
If the program does nothing to help Bob, then maybe a different keyboard or mouse would be worth trying, other than that I don't know what else to suggest. I'll keep the issue in mind though and if I hear of a solution I'll let you know.
Good luck!
Scott.