I disagree. Using 4H on dry pavement can lead to driveline wind up, even in a straight line. Depending on a lot of factors such as road conditions and tires, this can be a significant amount of stress on your driveshaft and axle. As JB posted, when cornering or taking tight turns, it can cause hopping and scrubbing at low speeds, anyone who has tried to park a truck in 4H knows what I am talking about. Another concern is driving at speed with very small differences in tire diameter due to slight variations in tire pressure or tire wear. According to some simple calculations (feel free to correct me if I missed something or made an error) at 45MPH a simple 1/16" difference in effective rolling radius between the front and rear axle can lead to about 1" of windup at the tire per second! Even more important is that this windup is magnified by 3.727 to the driveshaft, which is usually why the transfer case or driveshaft acts as the "fuse" and fails. Even if there is not a catastrophic failure, there is extra wear on all of the components within the powertrain system, which will decrease life of the system and lead to premature failure.
In reality, one of the axles will wind up to the point where the driveline gets strained until it has enough energy to break the tires loose while driving. At this point the driveshaft will unwind and then start the whole cycle over again. While driving on wet pavement or snow/ice, the strain energy to break the tires loose is significantly lower and within the design parameters of the 4WD system. If you want to get an idea how much effort this takes, go out and try to push your truck on dry pavement while it is parked with the brake on
I am fairly confident that some engineer across the pond went through similar calculations and continued to design the bearings, bolted joints and cross sections to handle these stresses. Since the manual states not to drive it in 4H on dry, I can safely assume they did not design it to handle those additional stresses. For short distances and slower speeds it is less harmful.
To answer the question of the OP, it won't hurt your transmission, but can hurt your transfer case or driveshaft. 4WD helps some on ice, but slowing down is the right answer. Switching between 2WD and 4H on the fly does introduce some system wear, but there are synchronizers that are designed to do this at or below the recommended speed. I would suggest some good winter tires, but I doubt they are worth it in TX and I doubt you could find a shop that stocks them!? Stay safe!
Assumptions
Diameter of P265/70R17 is 31.6"
µ*N > Net Tractive effort
tires are not slipping ( i.e. as in dry pavement)
Radius Delta of 1/16" for 2 PSI Tire Pressure Difference
this value changes significantally with different tires, but is very conservative
Df = Front Tire Diameter
Dr = Rear Tire Diameter
not necessary to know left to right tire diameters due to axle differentials (except for those of you running lockers..)
When travelling in a straight line in 4WH (transfer case locked)
π* Df = Cf effective ROLLING Circumference of Front wheels
π* Dr = Cr effective ROLLING Circumference of Rear wheels
assume front tires are low (if only one tire is low, reduce radius delta by half)
IF Df = Dr - .125"
Cf = Cr - .125"
0.125 Difference per revolution
when tires are not slipping and driving in straight line @45MPH
45 MPH
convert to inches per minute
792 Inches / second
99.27 inches / rev of tire
7.98 revs per second
multiply Difference per revolution BY revs per second
0.997 difference in inches / second