Go straight to the shop. Most OEMs will hire off of the streets, if you're willing to be under the national CBA. Right-to-work or non-right-to-work shouldn't matter in the aspect of getting into a Local. As closed shops are not legal.
In a right-to-work state you don't have to be in good standing w/your Local. In a non-right-to-work state you do have to be in good standings. I've Union in both kinds of states. If you're experienced, go into Carrier or Honeywell, ask if they can sponsor you as an MES Serviceman (if you can live w/that) & go from there.
638B here in NYC after th 45th day working for new guys you have to be placed in the union,and the training that follows....they sometimes ask you a trick question like "who's buried in Grants Tomb"..........."Who discoverd the Hudson River"
that,s funny- but not really a trick question, I think they just wanted to make sure you weren't a zombie.