Jeep tj 2006- when left sitting for few days engine will rev high - as if only operating on 3 cylinders - also does it when veh left idling on hot days with AC on. Most noticable in winter when sitting for few days. High rev sound but not necessary high RPM range!?

Fuse 18 or 19

Fuse Box Diagram Jeep Wrangler (TJ; 1997-2006)

https://fuse-box.info > jeep > jeep-wrangler-tj-1997-20...
The fuse panel is located behind the glove box. The location of the fuses in the passenger compartment: Jeep Wrangler (TJ; 1997. To access ...

1997-2006 Jeep Wrangler TJ Fuse Box Diagram

https://knigaproavto.ru > shemy > 920-1997-2006-jeep...
Oct. 8, 2019 - Fuse Box Diagram Jeep Wrangler engine: 2.4L, 2.5L and 4.0L (TJ; 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006)

Watch video.

https://www.youtube.com/watch?v=T3BcA9JEciU

how to fix a horn on the work bench
or how to diagnose a dead horn on car.?

my guess, #2
TJ car.
here is the list;

1. do not fiddle wires near horn sitting at steering wheel unless you defeat air bags first, per the BOOK "FSM" or u may die.
2. check the fuses first.
3. scan the PCM next, got DTC errors ?
4. read the FSM yet?

here, TJ
http://colorado4wheel.com/manuals/Jeep/TJ/

the pdf.
chapter 8H> horn
and the schematic there 8w?
8w41-3 see that relay, does it click with button pushed?
key on?
need more ask.
is ground good to horn
did you hot wire horn to see if its really good first
connector pulled?

I have an 05 6 cylinder automatic TJ. The other morning I was getting trouble codes U0101 and P1603. For a few seconds the Jeep was shaking and for the entire 30min ride it wouldn't shift into the last gear, it would just continue to rev. The next day I drove it and it was fine. I also noticed a slight high pitch whine noise coming from the engine at low speeds. Any suggestions? Thanks!

start with check fluid level --cooling fans operation --water pump to start

please say you did not get the engine steam cleaned...as you can cost yourself a LOT of money.hopefully the heat will dry things out, otherwise get the car put on a diagnostics machine.

Try removing the radiator cap when the engine is cool and top off the coolant level. Start the engine with the cap of and let it run to operating temperature which is when the thermostat opens. If you lightly grab the upper radiator hose. If it is cool or warm, the thermostat has not opened. If it is hot, the thermostat has opened. At that time have someone assist you by sitting in the car to rev the engine up (1800 to 2000rpm) and hold it there while you once again top off the radiator and tighten the cap. DO NOT BRING THE RPM DOWN TO IDLE WITHOUT HAVING THE CAP ON TIGHT! THE HOT COOLANT WILL OVERFLOW FROM THE RADIATOR! Then let the engine run to see if the air has escaped. You may have to do this procedure more than once. If at any time you feel that the engine is getting too hot, tighten the radiator cap and then shut the engine off. Then let it sit and repeat the steps above. Constantly monitor your temperature gauge at all times.

most likely a vaccum leak

TRY THIS.
Here are some of the causes of surges, stalls at stops, slow idle speed, erratic idle speed, rough idle and engine hesitation (and other problems), it is in most cases the idle speed control air-bypass valve and or throttle valve and upper intake, these area's get full of gunk and combustion residue over the miles and cause idle issues (stalls, low idle) like yours, Get a can of intake cleaner from any local parts store, not carb spray, intake cleaner, it is made by a company called CRC, remove the air intake hose to the engine, hold the idle high so the engine won't stall, then spray the can of cleaner into the intake while keeping the engine running, use at least 1/2 the can, shut down the engine and disconnect the battery for 5 minutes, then restart and complete a number of mixed driving cycles, town, freeway, stop and go etc., after a few days the problem will go away as the system will relearn to the clean intake.