I would have a FIT if somebody drove their golf cart on to my lawn--I don't blame you for being upset! You might need to talk to the golf club about THEIR liability for your property. Their signs may need to be a bit firmer (e.g., rather than telling players to stay off private property--the signs may need to say something like, "entering private property is criminal trespass. Do not attempt to retrieve balls on private property.")
We live on a pretty little creek that is the PRIVATE PROPERTY of the homeowners who border the creek (we each own halfway into the portion of the creek that borders our property).
I don't care if people want to walk in the creek and look for prehistoric shark's teeth (actually, I do care but I know I'll never be able to stop them all

) but I've had a few run-ins with people walking their dogs--not even in the creek bed, but in my yard!
To one particular person who couldn't get it through his head that he should not be walking his dog IN MY YARD (he actually said to me in a surly manner, "Well, what if I WANT to walk my dog here at 8 oclock in the morning?"), I replied, "Give me your address and I'll come over and walk my dog in your yard." This finally made enough of an impact to get him off my property. I have had to actually threaten a couple of people with calling the police.
For some reason, people who claim to be "neighbors" think they should have the freedom to traipse through my yard. We've even had parents with their kids & dogs who I've had to tell to stay away from my (fenced) dogs (yes, you are in my yard; no, my dogs are not friendly with other dogs; and no, they are not friendly to strangers trespassing in our yard).