Yeah, the Check Engine Light and different driving probably means you're getting forced into open loop or some other mode to protect the engine - I doubt it has anything to do with Tow/ Haul.
I'm guessing the truck is out of warranty since it's used? If so, I would try replacing the bank 2 O2 sensor (Get the actual OE/ OEM replacement and not the wire-it-yourself BS) and check for any leaks, cracked manifolds, etc. If it still pops up I would probably replace the injectors on that bank.