Q: My mechanic says my ford Explorer has a misfire but to check I have to wait until the check engine light comes on?

My mechanic had just replaced my car starter because it wouldn't start up. And my car has a new battery. My car has been stalling a little and feels as if its about to die out on me when I slow down. While reverse parking, I turn my wheels a lot more and my engine would shut off mid way but I can turn it on again and continue perfectly fine. When I am stopped at a stop sign or parked my car would vibrate. I had my mechanic test drive my car and he told me it looked like misfire or it could be something else but I have to wait for my check engine light to appear inorder to find the diagnostic. When will my check engine light come on? Won't it make it worse over time? My mechanic assured me that my car is safe to drive but he can't do anything until the light comes on while driving. And to keep driving it until it does. I'm just worried that I have to keep driving it and the light still never comes on, it's scary.
My car has an automatic transmission.