Once it warms up, it doesn't want to stay running once you restart it?

Hello,

It sounds like you might have an issue with the idle air control valve.
When the IAC valve becomes faulty it can impact the idling of the vehicle.
Sometimes if you have enough dirt/grime build up, it will not let correct amount of air in to the engine, the engine will have hard time idling when cold.
You might be able to clean out the IAC valve, which can improve the issue.

You might find these articles helpful:

https://www.2carpros.com/articles/how-a-idle-air-control-valve-works
https://www.2carpros.com/articles/idle-air-control-valve-service

If the internals of the valve is too corroded then you will have to replace it.

I have attached the diagrams for your reference.

Please let me know of any questions.

Thank you.