Steve-
Forget the California title and registration, it'll likely not be of any use to you, although I'd be sure to get copies of both signed over to you. States don't (usually) cross check VINs (Illinois can't seem to keep track of their own stuff, let alone CA's). Since you are using a different VIN (the one on the body) anyhow, whatever they were using will just disappear into disuse. You REALLY need to get a bill of sale (with the VIN from the Certificate of Origin) from whoever you bought the car from, and a Manufacturer's Certificate of Origin from Carey at Special Editions. Carey should be your first call.
Then follow the instructions on the link. There is no other way to be legal. Read the link. Re-read it. Memorize it. Start making calls. It's not that big of deal, but there are no short-cuts.
It takes as long as it takes. In 2005, there was only one lady that did this at the Secretary of State's office in Springfield, but she was nice and went pretty quickly.
The bad news is that it's not as straightforward as buying a used 'vette. The good news is that you'll not have to deal with the trolls at the local Secretary of State's branch.
You'll do fine. Read the link. Follow the instructions.