It depends on the tyres, the road conditions and the load. When I'm not towing, using AT tyres, I'll run 34-36psi. When I'm towing, I'll up the rears to 38-40psi. When I'm on loose surfaces briefly I'll leave them, but if I'm on really loose stuff or going to be on it for a while, I'll drop the tyres to around 24psi. For beach work you'd even consider going as low as 18psi.
But with that said, your tyres will have a recommended pressure on them, and that's because the manufacturer recommends that pressure for the build of tyre. If you are getting even tyre wear across the tread, your pressures are probably just fine.
What I am struggling with is the description of the issue where the road surface causes the wobble to commence. Caster angle is the first thing that comes to mind, because it governs how the wheel wants to sit on the road with respect to the axis of the steering and its motion in a particular direction.
The only other time I've seen something like this (experienced it myself, in fact) is where a light motorcycle uses a tyre like a Pirelli Phantom (not sure if these are still around, I use Pirelli Night Dragons now and my bike's not light). My little 250cc bike, fitted with a Phantom, would 'wobble' in time with the zig-zag of the centre tread pattern. It was made considerably worse with a firmer tyre. On a heavier bike, you didn't notice it. I really don't expect this to be the issue on a car especially with such a wide tyre, noting that they generally have a fairly uniform tread pattern - although that's for HT, MT and AT tyres. I'm not sure if LT tyres would be different in their handling.
So unless your power steering pump is on the verge of doing silly things and is triggered by the change in road surface, I'm running out of ideas.