Am in same boat as you. The only thing I could find that seems to fit the bill is through full tilt karting at the englishtown raceway. I am headed there Saturday to speak to them and go round in circles on their tag kart. They seem to run more than just arrive and drive services.
I will ask about racing leagues etc. for us older farts (I am 47) and will let you know what they have to say.
DOM – You should check out OVRP if it is close to you. I plan on going up Next Tues or Friday for Pratice. On weekdays, there is lite attendance so we get plenty of track time. It typicially ends up being open track sessions 10 AM to 6 PM). You may want to consider Renting a TAG Kart From them. If they are Okay with it, maybe you take a few laps in my Kart. I hope to race the following weekend in there club race, in Masters (Old Guy) Class. It will me my first Kart race in awhile.