Power steering drifts over uneven pavement

On a 2006, the towing weight probably split the Sway Bar bushings. The sway bar stops the back from swaying. Could also be shocks as well or both.

Push down on the back and see if it bounces.

When you turn at low speed you most likely had your foot on the brake. As such, you are loading up the front suspension i.e. weight shifted to the front. Backing up at low speed has the same effect of loading up the front suspension. Your front tires contact patch to the pavement also increase in area -- think about the tire is less round but more flat surface touching the pavement.
In both cases (low speed/foot on the brake and backing up), they put more stress on the steering and thus the steering pump.

Hopeful fixes:
1) check tire pressure of both front wheels especially the sound occurred more often when turning left than right. The correct tire pressure when cold reduces the resistance from turning the steering wheel
2) check and/or replace steering pump belt(s). When they are worn out, they became inefficient in helping the steering pump to provide needed pressure to assist steering. Now the steering pump has to work 'harder' to provide the needed assistance to the steering

3) your steering pump is wearing out. Not totally dead yet. In cold days, the steering fluid is 'thick' --think viscosity. It is less efficient in circulate the pump. As such, a 'weak' pump has to work harder to make the thick fluid to circulate (to provide steering pressure). Mechanics refer to this cold days steering noise as "Morning Sickness". You can try to mitigate the problem by changing the steering fluid. You might need to change it a few times before the fluid in the entire system is purged and completely replaced.

Hopefully, inflating the tires to the correct cold temperate; changing the belts and/or complete steering fluid flush could solve your problem. They are all much less expensive than replacing the steering pump.

not at same time ,pray tell.
on ice, dirt, pavement 2wd, 4wd.?what.
if on pavement and in 4wd, this is illegal actions by you.
use only on ice and dirt. in 4wd. unless car is AWD.

do not drive car if this happens in 2WD.
do not, its dangerous.
find a mech, now and find out what is bad.
can be many things.
damaged suspension or steering links.
bad wheel bearings
bad brakes
or other things..
even low tire pressure. it drops cold, , check it.

Check your fluid.

have you checked air inflation?Maybe more air up front will help you

check front end components, you may have a bad bushing in steering, or a bad part in front end. JUST GET UNDER AND CHECK IT OUT!!

My opinion is strain yes damage no. You know when you turn the wheel all the way to one side the tires are "grinding" into the pavement this causes you to give it a little extra gas to get the car to move...right? But I don't think that would cause damage. If I'm wrong then you have the worlds' worst engineered car!

Turning lock to lock as you describe is making the steering pump work hard since the wheels are stuck to the pavement.

I'm calling it B.S.

In a lot of GM and Chevys the intermediate steering link will lose grease and cause a clunk-like feeling while turning and braking. There is a TSB for such a problem. The dealer may still warranty the repair. You can take your truck into a local shop and they'll shake down the front end for about 1/2 hour labor to see if anything is wrong. About 80% of the time a pull is caused by a lateral tire pull or even a bad tire. Check your air pressures as well.

grass offers less resistance than pavement check level in powering steering pump/ and belt for correct tension

Front shocks are week