Yeah most clubs let you participate with a different car.
My advice to you is...get on
MotorsportReg.com : MotorsportReg.com > Online registration management tools for driving and social events and find events near you. Mid-Ohio is an amazing track, you are lucky to live where you do.
Find an event, get yourself an instructor (most clubs will force you to anyway) and have fun. I would think about taking care of the oil cooling problem first too. I am going to be testing a cooler soon, so if you can wait for my results before you make up your mind, I think you will benefit from doing so.
If you have any questions, don't be afraid to ask. We are all here to help.