If you disconnect both your chassis and house batteries overnight, the hard-reset might place Tow-Haul mode into "off" by default.
Make sure your trans fluid is at the proper level. If any is needed, MAKE SURE you add the right kind. Don't add just any generic trans fluid. All Ford trans require a specific trans fluid.
I would not worry about the situation because it sounds normal, especially after a cold start. Just take it out of Tow-Haul before putting the shifter in drive.
I wish our 2007 E350-V10 had your problem, Tow-Haul "on" by default. I say this because the weight of our rig with tow vehicle is in a Tow-Haul mode scenario 100% of the time, easily 15,000 pounds combined vehicle weight. Many years ago when our rig was a few years old, I called my local Ford dealer inquiring if they could program the Tow-Haul to be on by default. They said "no".
So....I put the trans in Tow-Haul mode right away, every dang time I remember.