2006 equinox high beams come on while using turn signal

Depends on the vehicle.
Lots of "modern" cars require that you press the turn signal switch away from yourself for high beam, and pull it towards yourself to switch to low beam. While on low beam, a gentle tug on the handle will flash your high beams.
It's a good idea to park facing a wall and then practice using the switches. you can see the result by looking at the wall.
Cheers.

PCB Relays
This vehicle is equipped with the following printed circuit board PCB mounted relays.
• High beam
• Low beam
• DRL
• Fog lamp
• Park lamp
• Stop lamp
• Horn relay
They are directly soldered to the underhood fuseblock circuit board. A failure of one of these relays will require the whole under hood fuse block to be replaced.

Turn Signal/Hazard Lamps
The body control module (BCM) is responsible for controlling the turn signals and hazard lamps. When the turn signal switch is placed in either the left turn or right turn position, the BCM receives a signal from the turn signal switch. The BCM then sends an on-off voltage signal to either the left or right turn signals and their instrument panel cluster (IPC) indicator. When the hazard switch is pressed, the BCM receives a signal from the hazard switch. The BCM then will send an on-off voltage signal to all the turn signal lamps. All the turn signal lamps will flash including both IPC turn signal indicators.

Vehicles are just not the same as they use to be. Electronics !!!!!

push down on the lever and the lights will go back to dim.

you'd have to play with it and see if it's related to maybe a stuck fog light or simply the high beam switch is junk

Your running lights which come on when you are in drive should be the ones flashing when you use the key FOB to lock or unlock vehicle not your head lights. Although unusual maybe both low beam head lights are burnt out or a bad ground, that the only thing that makes sense to me.

WAY TO GO !!! another "great idea" from vehicle manufactures

Have you tried wiggling the turn signal lever when the problem occurs? I have seen multiple cases where the wiring inside the headlight high beam/turn signal switch melts and causes the problem you are describing. If you can wiggle the high beam/turn signal switch and CAUSE the problem, you have a defective high beam/turn signal switch.

check relay and the fuse.

there should be seperate fuses for the high beams in the under hood fuse box try checking them