Hi Ed,
No pity needed - it's a good question.
You hit upon one of the essential basic functions of a solar charge controller - to stop charging once the batteries are full so as not to damage them. So yeah, unfortunately, you really need to have a solar charge controller.
Some of the other basic functions of a solar charge controller are:
* To prevent the batteries from discharging through the solar panels when there's no sun
* Optimize the charging algorithm so as to, in effect, pack as many electrons as possible on the battery plates.
And the more advanced charge controllers are smart enough to allow the solar panels to operate at their most optimal voltage (to squeak out the most power from them - up to 30% in some cases) and the batteries at whatever voltage they're at while they're charging up. This technology is referred to as Maximum Power Point Technology (MPPT).
You may have already come across these specialized solar charge controllers for electric vehicles/electric golf carts:
http://www.altestore.com/store/Charge-Controllers/Solar-Charge-Controllers/Electric-Vehicle-Golf-Cart-Chargers/c1213/They do actually use Maximum Power Point Technology and, as you might've noticed, make it particularly easy for you to use solar by allowing you to use a normal 12V solar panel and then it up-converts that automatically to match your 36V battery bank in the golf cart. Otherwise, you would need to buy 3 x 12V solar panels (which means smaller panels and probably more expensive per watt than a single, larger 12V solar panel), put them in series for 36V and buy a charge controller that's rated for 36V in and out. Overall this 3x12V solar panels solution would be more expensive and more complicated to implement.
Hope that helps!
PS: For more details on how solar charge controllers work, differences between traditional (PWM) solar charge controllers and the ones that support MPPT check out these articles on the Learn section of our website:
How Charge Controllers Work
http://www.altestore.com/howto/Solar-Electric-Power/Design-Components/How-Charge-Controllers-Work/a3/How MPPT Charge Controllers Work
http://www.altestore.com/howto/Solar-Electric-Power/Design-Components/How-MPPT-Charge-Controllers-Work/a13/Comparing PWM and MPPT Charge Controllers
http://www.altestore.com/howto/Solar-Electric-Power/Design-Components/Comparing-PWM-and-MPPT-Charge-Controllers/a100/Be aware that these articles talk mostly about situations where the battery banks and solar panel voltages are the same or where the solar panels have a higher voltage than the battery bank. In your situation the solar panels have a lower voltage - but that's ok, nearly all of what's written is applicable.
And, of course, you can always call the toll free number to get help from one of the knowledgeable technical sales team folks.