Update -- 90% there...

After the acid bath treatment, I reinstalled the dometic and fired it
up on LPG.  Ambient temperature in the mid 70s.  Within 2 hours it was
below 50, and within 6 hours it had made ice cubes and was down to
36F.   I don't think it's ever worked that well before.

At that point I turned down the thermostat to about 1/2 way, and went
to bed.  The night was still, very little wind, ambient temps dropped
to about 60.   Woke up this AM and the flame had gone out, but it lit
right back up again.    Maybe turning it down from "max" was pushing
my luck?

I wanted to again thank all of your for the fast responses & great
advice.  This list is an amazing resource :-)

I'll leave it on max and see if it goes out again.   Any thoughts as
to why it'd go out overnight when not set on max?  Perhaps the "low"
flame size is too small, or perhaps the thermocouple is not getting
enough heat on low?  Does "max" force the flame size to "high" at all
times, or is "max" just the highest thermostat setting (meaning the
flame size can still go to "low" if it gets cold enough?)