You may be able to gift the cars to each other and pay a lower gift tax, if there is one available in your states.
Because they are from different states and because it becomes a private sale no you will pay tax.I have to ask why dont you want to pay the sales tax that goes to help pay for schoo;s and such?
By having you pay tax on market value rather than actual sale price, states have ensured that they will get their take. It is not technically a sales tax but a registration fee which should be deductible on your federal income taxes.
Look into what sort of documentation would be needed to show the transaction as a gift and you may be able to lower your liability.