Seeking advice on following situation. I have a PLASTIC u-tube (TRAP)
connecting
the downspout of my kitchen sink, to the next pipe in the sequence.

I have replaced the gasket and applied TFE Paste to the male part of the
tubing that screws in to the female connector, tightened it as tight as I
dare, and yet it still leaks.

If your trap and piping is plastic and is still leaking, you re using the
wrong gasket or have it in backwards..it should be a tapered plastic ring
(the color of a milk jug) the narrow edge pointing "downstream" . Its made
to work by turning the coupling HAND TIGHT, without ANY teflon tape or
paste. If your "next pipe in sequence" is metal, it would probably be
easier to replace the whole leaking metal set up with plastic than to chase
the leak around . Make note of exactly what you have at the wall, maybe even
take a digital pic of it and go see yer plumbing guy at Lowes or Home depot.
The parts should be under $ 5.00