Glad you liked it!
I think the term “warrior order” wouldn’t really work for either kingdom. The North doesn’t go in for formal institutions – compare Robb’s companions to the Kingsguard with its elaborate oaths, uniforms, etc., or the Old Gods to the Faith of the Seven – so the North’s “sworn swords” are about as formal as it gets.
Similarly, the closest the Ironborn get to specializing in that way is the Iron Fleet.