When towing with my 2004 durango in tow haul mode

Tow/haul mode enables the operator to achieve enhanced shift performance when towing or hauling a load. When tow/haul mode is selected, the tow/haul switch input signal to the body control module (BCM) is momentarily toggled to zero volts. This signals the powertrain control module (PCM) to extend the length of time between upshifts and increase transmission line pressure. Cycling the tow/haul switch again disables tow/haul mode and returns the transmission to a normal shift pattern.

Would't trust it , take an have DTC'S - diagnostic trouble codes checked by a qualified repair shop with a factory or professional scan tool . Auto part stores can not check BCM code's .
DTC B2722

Tow/Haul Switch/Indicator Always On or Inoperative

Step
Action
Value(s)
Yes
No
1
Install a Scan Tool .
Turn ON the ignition, with the engine OFF.
Cycle the tow/haul switch while observing Tow/Haul Mode on the Scan Tool .
Does the Scan Tool display Active when the switch is first pressed, and then Inactive when the switch is pressed again?
--
Go to Intermittent Conditions in Engine Controls - 4.3 L or Intermittent Conditions in Engine Controls - 4.8 L, 5.3 L and 6.0 L
Go to Step 2

are you shifting into a lower gear when starting to go up the hill you should use the transmission when doing this no matter how steep the hill is because this will stop the overload on the transmission , maybe time for tune up ,? , do u also have a towing switch in you vechil I do not know what u are towing with , no listed in your question

have the tires checked for belt separation or out of round
code p 0300 is random misfires and not cylinder specific
code p 0301 is for no 1 cylinder
check fuel pressures and no 1 injector for problems as it is more more fuel system problem than plugs

That is the Tow/Haul indicator. Button on the end of shift selector was most likely depressed during shift or possibly a malfunction with the tow/haul mode selection system. (IE: Tow/Haul button)

Where is the question this appears to be a statement of an answer to another previous question

could be the switch or modual.check wiring for worn through or earthing.

whens the last time u changed the fuel filter???????

tow mode gives ur transmission a longer shift point to relive strain on engine while driving -it doesnt give it more horse power to start off a hill

You should as it also manges the control system for the trucks ABS pump as well as the transmission line pressures.

If you are "lugging" the engine, you will get increased exhaust temperature (EGT), this will trigger your "turbo" light(not a good thing), ideally you should have an aftermarket "pyrometer" fitted to your car, this will show you the optimum revs etc you should be using in any given circumstance.Or revise your driving technique, ie change down gears earlier.
The turbo should be idled down after long trips or heavy towing, this allows the oil temps to stabilise and turbine speeds to stabilise, thus idle for 2-5 minutes.