Vehicle is running cold, A/C hot and engine light on

Sounds like we may have two different issues. I doubt A/C being hot is causing the engine to run cold.

When you say it is running cold, what is the temperature that it is running?

Also, we need to check the codes and find out what are there.

https://www.2carpros.com/articles/checking-a-service-engine-soon-or-check-engine-light-on-or-flashing

Then we need to check the pressures of the A/C system but I suspect we are going to have to start a new post for that issue as I doubt it is related.

https://www.2carpros.com/articles/re-charge-an-air-conditioner-system

Let's start with the check engine light and then figure out what the actual engine temperature is and we can go from there.

Thanks