The blinkers are run by the BCM which sounds like is having a problem. Usually the blinkers will turn off with the key if it was something like a switch but the BCM stays live with the key off.
To find out for sure we must scan the CAN here is video to show you how.
https://youtu.be/InIlnsjOVFA
If it is the BCM you will need to have a shop program it for you it should cost about $130.00
Check out the diagrams (Below). Please let us know what you find. We are interested to see what it is.
Images (Click to make bigger)
Thursday, September 21st, 2017 AT 4:22 PM