Tail Light Wiring Issues

When you say " confirmed functions with multimeter", do you mean that you checked Voltage? If you're getting voltage, the system is functioning properly, the lights aren't. How are these grounded? You don't mention a ground wire. The factory lights are grounded when screwed to the body. If the same, you need to make sure that you have a good ground when testing. If not, make sure that the ground wire is connected solidly.
When you make your final connections, use crimp on connectors. I like to use UNINSULATED connectors with heat-shrink over them. Use a GOOD crimper, not the cheap junk you can get everywhere. If you think that someday you might need to remove them for some reason, insulated spade connectors work well.
Yes tested all things functioning correctly.

Of note, the lights and light casings themselves are dangling out so perhaps it isn’t grounded yet? I just didn’t want to screw everything back in if they weren’t working right.
 
Did you ever figure this out?

No, I tried wiring it several different ways. After installing a new set of LED tail lights I was able to get the passenger side working but the driver side would only have the normal red lights not the bright brake lights or reverse. Since the issue seems to be with the turn signals I'm wondering if I need to replace the turn signal relay?
 
No, I tried wiring it several different ways. After installing a new set of LED tail lights I was able to get the passenger side working but the driver side would only have the normal red lights not the bright brake lights or reverse. Since the issue seems to be with the turn signals I'm wondering if I need to replace the turn signal relay?

If you have the right side working. Get a test light and check the circuits see if you have power when the brake pedal is depressed. You may have to follow the wires to see if a repair was made at some point and different color wires were use.