#5 (5A) keeps blowing

That fuse powers more than just the dash lights as the diagram shows. I suspect you have as issue in one of those items causing a short and blowing the fuse as a result. At the lower settings the dimmer absorbs a lot of the current while turning it up causes the failure.
Probably the easiest way to find the problem will be to go to the junction connector 4 at the left side of the dash and unplug it. Then see if the fuse fails. If it does then check the shift bezel light circuit for the short.
The others will be a bit harder to find as you will need a way to measure the current sent into the various circuits.
Does it have the stock radio in it? If not, that could be the problem if it shorted in the radio?
The only other way to test would be to unplug the parts listed and see if the short goes away. Then trace the circuit that causes the fuse to blow.

Unplug it and see if the fuse stops blowing.

The switch is getting hot because of the short that is blowing the fuse. The dimmer for those lights is nothing but a resistor in the headlight switch. Until the short is repaired it will get hot as it normally only carries the current that keeps the lights lit up. With the short, however, it is trying to protect the fuse from blowing and sinking the extra current due to the short. Finding the wiring fault will take care of both problems.
Start with the easy things, it's fairly common for radios to develop faults in the wiring, especially aftermarket units so I suggest starting there. Then the ash tray lamp, then the HVAC controller and finally the instrument cluster. Unplug them looking for the higher current draw. It's also possible that the wiring itself rubbed on a bracket or similar and is shorting there.

If you can live without the lights in the dash, then leave it out. The A/C, radio, dash, and ash tray lights should be out with the fuse out. If they are not, you have a different issue where they are getting power from somewhere else. However, if they are out then you still have a short in one of them. If unplugging the radio, not just looking at it but actually unplugged, and the issue is still there then you need to look at the other light feeds. Removing the fuse and having the switch cool down just confirmed it is in the light circuit the dimmer feeds. But it's your call.

If that's the case, then you have what we call a thief circuit. There is a short between two different power sources or there is a crossed wire. Do those lights go off with the key or the light switch?

Okay, the video shows an obvious problem. The lights on your dash are not the OE ones, they are some LED strip lights added later, The OE lights come on behind the gauges and the light goes through them. My guess at this point is that someone tried to connect the LEDs and the radio up to the factory dimmer and damaged the dimmer circuit from high draw.
Now if you added that strip it is a different issue. Please advise.

Okay, I was hoping that was the case but wanted to check. What make and model is the radio? Did the lights go out recently or were they out when you first got the truck?
Trying to narrow down what may be the issue. I have seen similar issues when the light feed for a stereo was connected to the dimmer feed from a car as it tends to mess things up. Before the switch burnt out did the lights in the shifter, ac and ash tray work? Could they be dimmed? You said it doesn't have the ash tray light, I wonder if it does, but it fell out and is now shorted somewhere behind the dash. I wish this truck was here, it would take about an hour to sort out the problem, just hard to explain the thoughts behind the testing. See if you can find the joint connector 4 I attached the diagram for, it is behind the dash on the left side, what it does is act as a bridge connector, with it unplugged it would be easier to test the other circuits.

You shouldn't have the shifter lamp installed, but the wiring may still be there. The ash tray lamp is usually in the plastic above it so that when it's open it lights it up.
Looking at the info they have to check to see if the radio has the orange with white stripe wire connected to an orange wire in the truck (or if any wire is connected to an orange wire). If it is, try disconnecting it and see if the other lights now work.