Engine will not stop when key turned off

Yes, this is a classic GM problem, here are the fixes in order of popularity.

1. Faulty Ignition Switch (very common on Equinox/Terrain)

2. Failing Body Control Module (BCM) - Common on this era of Equinox

3. Stuck / Backfeeding Alternator Field Circuit

You might want to do a CAN scan which may give us code to help us fix it as well, check out this video below.

CAN SCAN - https://youtu.be/u-4syLc-ifQ

If it is the alternator you can disconnect the main charge wire and isolate it so it doesn't touch anything and then start the car to see if the problem goes away. Here is how to change out the ignition switch which is usually the problem. I will email you the images as well.