When I buy vehicles from private parties, I use a check from my credit union.
I apply for a used car loan (even if I have cash but don't want to carry it). When it's approved, they send me a check with a "not to exceed" amount printed on it, and a place for me to write in the actual cost. Along with that, they send a paper explaining the process & their contact info.
Once I decide on a vehicle, I give that 2nd sheet to the owner, explain it all to them, and encourage them to call my credit union, give them the check number, sale amount, etc., so that the credit union can verify that the check is real & not a scam. Then we go to the bank, get the title notarized, I sign over the check & they can deposit it (if we're at their bank) or cash it (if we're at my bank).
When I've sold vehicles to private parties who want to pay with a bank check, I like to do the transaction at that bank so they verify that it's legitimate. If that doesn't work out for what ever reason, I'll minimally ask for a photo of the check (at least the bank its drawn on, the date, amount, & check number). Then I call the bank and verify that it is a good check.
As for driving it once the title is signed over, that's going to vary from state to state. In AZ, I print out a 3-day limited use permit the day that the transaction takes place. It costs about $1, & if IIRC, all I need is the VIN & my name/address/phone. Once it's signed over, I hang the temp in the back window, & I have 3 days to get the emissions test done & get down to the MVD.