Yeah, I hope there are more locations to check an alignment than that place! Either pretty unsophisticated equipment or they didn't want to help you. Never heard of a machine that couldn't check an older rig, but they are getting pretty high-tech and dedicated these days, so maybe that's normal.
It's hard to check truck linkages for looseness with the tires off the ground believe it or not. Even when they're well worn, they resist movement by anyone but the strongest strongman with a prybar! But since you say they're fairly new, then they should be fine like you said.
But for the full test, leave the full weight of the rig on the ground and have a helper saw the wheel back and forth about a half-turn or so, to make the tires squirm on the ground. While that is going on, closely inspect every joint and junction and nut and bolt for any flex, cracks, spinning, or side-to-side movement that is not directly moving the wheels.
It's the easiest and quickest method to check every linkage and even the frame behind the box. A crack you never knew existed could be sitting their completely closed and hidden until the torque of the steering works against the weight of the truck and friction of the tires, and opens up clear as day.
Any suggestions to what could be causing the very sensitive steering response?
As far as what could be making it darty, it's STILL the tires. Your test tires might have been fine, but you then had completely mis-matched tires between front and rear. And that one thing ALONE could make an otherwise good driving truck completely uncontrollable and dangerous under the right circumstances.
If you want to be sure, swap all four tires and see how it drives. I bet it changes it's manners drastically.
Paul