I bought a whole new connection pipe and installed it. I went with PVC.
(The cracked nut on the old connection pipe was permanently attached. )

The new connection pipe is working. HOWEVER, I've been checking for TINY
leaks by sticking edges of paper towels around the edges of the nut at the
connection. KEY QUESTION: Should I have ZERO tolerance for the paper towel
showing any sign of wetness when I do this or is a slight bit of wetness
considered acceptable. I've already gone beyond hand tightening, and I
don't want to wind up cracking the nut and having the same problem as
before.

(by the way I did use a new plastic washer that is the same type used as
before. It's the kind that has a flat edge that touches the sink's-end of
the sink's connection)

Thanks,

J.