Am I responsible for transportation

Am I responsible for providing transportation for my wife after she has moved out?
Both vehicles are in my name. One is a gift; one I bought and paid for entirely.
I have made all the repairs on both and except for her doing an oil change recently
(I only made her do it because she kept saying for weeks that her schedule wouldn’t
allow me to pick it up to do one). She laughed and said if she blew the engine,
I would have to get her another car. We both have made about the same amount
each for the last 10 yrs.

I was thinking about signing it over to her, if that won’t come back to bite me
when we do agree to separation agreement (which will probably be a while).
If I did that, she could pay for ins, tags, taxes, repairs.

Am I responsible for furnishing her transportation at all? There are no children involved.

You didn’t say when you paid for the car. Even if it is in your name, if it was paid for during the marriage with marital funds, it is marital.

No, you will not be required to buy her another car if one of the cars dies, but I wouldn’t take a car from her if you hope to settle this matter amicably. I would suggest that she begin covering the expenses for the vehicle she drives. Don’t sign it over until you have a signed separation agreement.