Nope, you won't both be charged. Just you.
The charge is driving without valid plates/registration and insurance (which is two charges), not selling. I don't think there's a charge for selling in any jurisdiction, unless the vehicle being sold is stolen.
You know that thing you said where you just bought the car? That defence doesn't count for squat. You can buy a car any time you want from any seller you want, but you can't drive it on public roads without meeting the legal requirements. You're supposed to line up insurance and tags before you drive it, no grace periods and no excuses.
Everyone else is expected to know that, which means so are you.