Doubt the tires are the cause, but could be some of the effect.
Either way, almost 20 years old, 130k miles, HEAVY front end = you could be due for more front end work. Anything loose, rode ends, drag link, Pittman, steering box (did you put a brace on it?) steering damper, could be the cause or part of the cause.
Or if it's done it forever, could be not enough caster in the front axle causing it to wander. A lot of Rams are indexed to around 3 degrees of caster. Upping that helps with road wander.
Other part, is, well, they just don't handle awesome when you drop 2 tons in the bed.
I'd suggest finding a good front end shop so you're not just throwing parts at it and have them check it out.
