Hi Everyone: Thanks for the replies, they were helpful. It makes sense what "Hey There" explained about the tow/haul mode in my 1500 GMC - that it raises the RPM of all the shift points where the transmission shifts gears. In hilly country I found using the tow/haul mode beneficial as the transmission shifted less and lugged the engine less resulting in slightly lower engine and transmission temperatures. On some long (>1 km) hills, steeper than 10%, I found out I had to shift the truck into 2nd gear and in some cases 1st gear to help hold the load back so I did not have to ride the brakes so often and over heat them. Down shifting early in combination with using the tow/haul mode was effective. Fortunately most of these steep grades also had sharp turns where you did not want to be going faster than 30kms/hr. so this was not an inconvenience to other descending traffic.
In regard to your "c" clamp suggestion, I did install a roadmaster anti-raddle devise on the 2" receiver before leaving on the trip and I can attest to it removing all of the flopping you described (looks like a big square "U" bolt).