Under Dash Phantom Heat

It could be the exhaust. My guess is, if you got a 5 speed, it's the tranny. Mine did that until I got a metal cover under my shifter boot.

I'm not sure you can check the cats without taking the exhaust off, which can be annoying. I'd do all the easy and inexpensive stuff first. Not too sure about the 04, but you should be able to pop off the shifter cover and check this little shifter boot. Not 100% put it fixed my problem.

https://lmr.com/item/LRS-7277E/79-04-Mustang-Lower-Shift-Boot-With-Metal-Ring

Me I'm a practical guy. As such, +1 on above. Do the cheap and easy stuff first.

As for the possibility of bad cats. How about just take it to a muffler shop and ask if there's a problem? You don't have to have a repair done. Also note, that IF the cats are clogged likely there will be other symptoms. Such as lower gas mileage, Reduced power.

It seems this could also be a heater blend door that's not closing all the way. Does it make a difference if the temperature control is moved? Trying to figure out if the control is actually moving the blend door and if the heat is coming from the heater or some where else.

Another option is to temporarily loop the heater lines in the engine bay. Then if the heat goes away this will confirm the source.

Note, on the 4.6 the heater flow is needed. As such it's a bad idea to bypass the heater core by blocking the lines. It's OK for a short term test.

Another thought, I notice on a lot of Mustangs that the accelerator pedal is close to the transmission tunnel and, over time, a hole can wear in the carpet and insulation letting more heat into the area.