A/C not working in the CAB

From your description it sounds like a stuck blend door. The mode selector sounds like it's okay as you can send air out the floor, dash and defrost. The cold lines should mean the actual A/C system is working, you just are not getting the cold air through the HVAC doors. On that truck it should be a manual cable control for hot and cold. However it isn't fun to get at the cable on the door end, you have to remove the dash to get to the clip that holds the cable to the door crank.
However you might want to pull the heater control out first just to make sure that the cable didn't fail on that end first. To do that you remove the dash trim then the 4 screws that hold the control in place. Then pull the control out and you should see the clip that holds the cable in place. Try moving the control and see if the cable moves. If it does the other end has either broken or more common it has slid in the clip.

Wish it was something easier. While you have it out you may want to look at the vacuum lines for replacement if they show rubber rot. I would also look at the seals on the HVAC module and check the cores for any issues.

Only one door. Hard to see in the images but it looks like the lever is moving, if that is the case but you still don't get cold air then it may be the door itself is bad or something got in between the door and evaporator and is blocking the airflow. That would mean you need to open the HVAC housing itself to check the door itself, or slip a camera into the housing and use it to look at the door and the cores.

True, but you may only be hearing a section of the door actually moving. If the top quarter is still OK it would act like the entire door was working, but it actually isn't. If you can get in there and look you may find the answer. It's also possible that something got in there and blocked the air. I repaired one once that a swarm of insects had blocked. That was a mess.
Let us know what you find.

Well, that looks like the entire door. Air should be moving across the core with that. Verify that it's fully closed but it looks like it is. To do that you rotate the temp to full cold, then move the door lever so it's at full cold. Once set that is all that controls hot and cold on them. I don't recall if you can run the blower with it apart, but you might want to run a set of jumper leads over to the blower motor and double check that the air is going through the cores properly. I don't really see anything wrong in that video but if the core is getting cold the air should get cold as well.

Possible but the system normally wouldn't get the lines cold if that was the case as it tends to restrict flow and you wouldn't get the signs you have with the condensation and cold on both lines. One of the trades with used vehicles is you don't know the full real service history of them. Did they say anything about replacing a compressor or drier?

Hmm, sounds like a bad ground connection. Possibly ground 202 on the left side of the instrument panel. It is shared by the dash and the window switches. The blower has a ground on the right side of the dash. I would check those first. Chrysler has a thing for bad grounds, the terminals tend to fail at the crimp. They are all black wires that ground those items.

The earlier Jeeps had something like 6 main grounds on each side of the dash, when they started to fail things would go crazy on them.

So you are back to the starting point. At least you know what it isn't, and by repairing the grounds you likely are preventing other failures. You could throw a set of gauges on it and see what the pressures are but your earlier description sounds like it is working somewhat. Just odd that the lines are cold and causing condensation but the air over the core isn't getting cold. About the only thing I can think of would be to recover the charge in it and recharge by weight and test it and see if the compressor is weak or possibly there is a blockage in the system.