How do you access these parts? You pull the steering wheel, and all the mounting bolts are accessible from the inside. Each part slides off the steering shaft. The first layer behind the wheel should be your horn ring, next should be the multi function (turn signal) lever, after should be the hazard switch itself.
The control arm not staying up for right turns makes me thing the little plastic ticker thing in there is either broken or too worn down to stay locked. Once you get the wheel off, you should see exactly what I'm talking about.
If you've ever worked on turn signal switches before, this is no different...other than all the crap you go through to get to it.
As BadassBronco said, the turn signal and hazards run off different flasher relays. You'd want to locate them under the dash near the top of the brake pedal and see which one isnt working as a possible replacement. Or, it could be just the switch itself has dirty contacts and needs to be cleaned.
Check the relay first, if thats no go, pull your steering wheel to fix your combo lever and check the switch too while you're in there.
Or.... you could move out of NJ to a state that doesnt require inspections. :toothless