My truck wandered so badly it was like “trying to herd cattle down the road” (a phrase I stole from HRL) and my truck came with radials. It wasn’t until I fixed a worn spring mount, rebuilt the springs, replaced the king pins, tie rod ends and adjusted the steering box that I got her tamed down. I’m not saying you’ll have to replace everything that I did, but any one of those items can cause your truck to wander.
Now bias ply tires as you probably know will pull you from side to side when crossing inline cracks, but they shouldn’t cause you to wander.
Jack up the front of your truck and move things around (wheels, tie rods, steering box, etc.), and see if anything has any play. Also wheel alignment plays into it. Read about adjusting your steering box. It’s a fairly easy procedure and can cause the wandering your talking about.
Here is a link on adjusting your steering box.http://chevy.oldcarmanualproject.com/shop/1942_47/4247csm901.htm