I agree with a short in the steering column, especially if everything works when you mess the signal switch. It could also be a bad BCM, but I would start with the lever. A T25 torx will remove the bezels around the switch, and the switch itself. Inspect all the wiring. I have like 3 or 4 signal switches if you end up needing one.
As for the airbag light, check the sensor connections.
1. There is one sort of under the right headlight/turn signal/side marker. Make sure the plug is tight. I would simply unplug/plug back in.
2. The main control module is under your center console next to the E-brake. It will be obvious to you when you see it. It's the ONLY module under the center console next to the E-brake.
3. The last and possibly most likely culprit (short of a simple 'battery' reset) is the clock spring, located directly behind the airbag on the steering wheel.