I'd check your wiring by the coils. From moving the wire harness you could of cracked a wire or check and make sure your harness plugs are snapped in all the way.
My brand new one moves as well. Not quite as easy as yours but it does move. Wondering might be caused by tire pressure being too high or too low, toe, axel not centered and/or caster can cause wondering.
I had a headlight bulb do this years ago in my Ford Explorer. The bulb somehow malfunctioned and had both high and low beam on at the same time. It started to get so hot that it melted the headlight housing, the bulb body, the socket where the bulb connected to the housing and my OEM wire...
Installed new Auxbeam headlights. Love these! They're bright with a nice cutoff. The halos and the startup rings are bright too. I'll try to get a better startup video.
Did you remember to tell the Jeep you switched to all LED lights? I was just thinking and remembered that when I switched to LED I used JSCAN to program the Jeep from Halogen to LED.
I doubt they are left and right sided. It's very possible the manufacturer messed this up. It happens more and more. You might be able to get a thin paperclip and pop the pins out of the connectors to swap the wires if you're not comfortable splicing the wires correctly.
My wife's friend has an 18 JK but found out there are some JL features installed too. Her JK has the auxiliary start battery that the JL came with. She was having starting issues. The dealer installed a new battery, programmed new fobs, flashed the security, and it still had the same problem...
Welcome to the forum! Get rid of those junk terminals. Wiggling the blue fusible link alternator wire might of wiggled that terminal enough to start it. I had no start issues. Thought the terminals were tight because all the nuts were tight but the terminals weren't tight enough to the battery...
Did you scan it and see what code(s) are stored for the HVAC? The light blinking can mean a few things like blower motor resistor is bad/corroded, rear defrost relay/fuse is bad, actuator(s) need calibration, or the HVAC unit itself is bad.