That’s a whole lot to respond to, but in short: if you want to lead you shouldn’t be letting another TM push you around on tasks against your TLs wishes. You don’t have to be a dick about it, just tell him/her that the TL/LOD/other leader said to do “x”, so you’re going to do x. Let’s get x done, THEN circle back to “y” task.
Conflict resolution is something to work on and the situation above could help, but really you need to be able to step aside personal conflict and be able to put business needs first. Filling new freight and ours is more important than a freezer hunt. Don’t get tied up in other TMs B.S.; if they don’t want to follow direction, leave them to be performanced out.
Honestly, PA’s to me have no true authority over other TMs at all (maybe it’s different at a SuperT); therefore, I’d respectfully tell the dictator PA you’ll stick to your assigned task(s) and proceed about your business. If the PA gets shitty, that’s reflective of them, not you.