I'm going to make many assumptions on this issue.
I'm going to assume you are pulling the temperature from the OBD2 port, not the cooler line (hotter), and not the pan (cooler).
I'm going to assume that you do have the T/H enabled.
For all other replies; the '05 transmission T/H mode does NOT lock out overdrive. It does lock the TC in 3rd and at a lower speed in 4th.
freddmc, at what speed are you towing? If you are in hilly winding terrain if you slow down slightly the trans will hold 3rd longer. If you are powering hard out of a turn the TC will wait longer to lock up in 3rd, just ease up to about 20% throttle position then get back on it and the TC will hold.
But with all that being said, I doubt it is your driving style. I'm betting that your cooler thermostat is stuck in bypass mode and causing the warmer temps (especially since it is currently winter time).
Pop the hood and you have easy access to the secondary cooler (which is not part of the radiator). Read the very first post in the link below about how to access and remove the thermostat.
CumminsForum 3rd Gen Trans T-Stat