Very cool, congrats on diving into something you've always wanted to do!
I know it may not be as thrilling as driving one of those dragsters, but why not get yourself a low-budget muscle car and hit up your local track more often? I bet for under $3-4K I could get/build something that would at the very least be fun, safe and not totally embarrassing going down the track. It doesn't need to be all that fast to be fun.
Back in the day my buddy had an 84 Z28 with all the typical bolt-ons, it was a rough car and looked like a POS but he used to take it to our local drag strip every other weekend and run the crap out of it. Loads of fun without breaking the bank.