Update:
This is crazy, but I think the welding worked!
I just tacked each cap on in two spots. I drove all the way to the boat ramp and all the way home. Which is only about 8 miles each way (16 total). But previously, both caps would start to back out after just ONE mile.
AND both hubs were cool to the touch. So I can't really make any sense out of why this happened in the first place, after 30 years of problem free operation. But its nice to be able to drive around without any stress again.