Typically the sales tax is paid where you register the car, regardless of where you bought it. At least that's how it is in MO. Wouldn't save anything.
My situation is odd and confusing. I still own a condo in MI where a relative lives and I'm renting down here. TN law says I need to get a TN drivers license within 30 days but I don't know how they would know if I didn't, which I haven't, until I go to renew in Jan. I don't see any reason to surrender my MI license unless I get pulled over between now and then. Livin' on the edge!