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I would check the steering gear box, if the pump was going it would be hard to stop and steer. Sounds like something has some excessive slop, how are the tires wearing?
Did it by chance start when you got the tires?
You need two people, your looking for slop, start with the steering gearbox, if that looks OK, check all the steering joints.
With the truck not running move the steering wheel back and forth, but your only looking for slop, once you feel pressure stop and turn back the other way. Now your always going to have a little slop, say a inch or so, if you have more you need someone to turn the steering wheel back and forth in that slop area while you go looking for the slop. Start with the steering gearbox and work yourself along all the joints until you find the issue.
You might not have any looseness which would mean maybe it's time to have your alignment checked.
Let us know what you find out.
If the new tires have weaker side walls - taller tires, lower load rating, etc. they can definitely feel like they are wandering compared to the previous tires. It's also not unheard of to have a defective tire, even new.
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