Potentially upside down on my car

Hello everyone, looking for some basic advice on my car and what I can do about my loan.

3 years ago I gave birth to my second child and had to purchase a new vehicle. At the time I was driving a Prius C and it quite literally could not fit two car seats in the back seat. It felt like the best choice at the time to buy a van. I was not as financially smart at the time and the market was very high.

I purchased a 2020 Chrysler Voyager. Financed at ~7% interest at 28k. I ended up buying a warranty for ~3k, which really saved my ass in more than one way as the transmission has gone out twice now and has been covered by the warranty.

Currently the car has ~140k miles on it and I’m experiencing a coolant issue that points to a head gasket issue. To fix this issue will be both costly and frustrating as this car has been full of problems.

I am considering trading it in for something more reliable, but the value of the car is less than what I have left on the loan. I am at a weird spot in life as I am about to start student teaching and finish school. I can’t afford to get a second job because time (full time student, working full time, 2 kids, etc). I currently owe right around 12k on the car. Edmunds valued it for trade in at 8.5k and KBB valued it around 6k.

Either way, I will be upside down on my loan if I were to trade it in and finance a new vehicle. I know I could get a new loan and roll it into that one. My FICO score is high 700’s, I think it’s rough 778. My question isn’t whether I could do this, but whether it’s the best option.

Another option I have is to lease a car through my partners work. They offer a good discount to employees, and this is something I have considered, but I’m curious how my current upside down balance would roll into a lease. Is that a thing? Would I be better off selling my car to the dealership or privately and getting a personal loan to cover the difference? KBB valued my private sale value to be ~9k and Edmunds put it just over 10k.

Just curious what the community thinks would be my best option! If it makes any difference I don’t have very much other personal debt, I don’t currently have any student loans outside of ~3k for a previous degree that is currently deferred.