Clay's logic is sound for many games, however I don't think there are any targets on GP that toggle between 1000 and 10,000 pts.
I suggest checking 2 other possibilities:
1) There is a 10-prong Jones plug that plugs into the lower left corner of the head (FYI: In the manual, Williams calls the component board in the head the "insert"). In that plug, there are 2 adjacent prongs/sockets that go to the 10,000 and 1000-pt relays. Inspect both the male and female ends of that plug to make sure there is no metallic debris or exposed wire that might be shorting those 2 prongs together. A short at this Jones plug could result in your issue.
2) A consultation with the schematic (area E-20/21) indicates that the #1 star relay indeed has a make/break (M/B) switch where one pole goes to the 1000-pt relay and the other to the 10,000-pt relay. This could also cause your issue and is similar to what Clay was describing as a misadjusted M/B switch where the center blade is actually touching both outer contacts.
Please check these 2 possibilities and report back. If it isn't one of those, it may be a more convoluted problem.