• Fixing this is the right way, i.e., taking the ceiling tile out,
replacing the drywall with cement board
and replacing with new tile, is beyond my $$ means, as well beyond my
own skill level. I'll just have
to settle for redoing the grout.

It is quite simple and cheap to just fix the ceiling. If you can re-grout
you can install tiles. A sheet of cement board is under $20.00, screws
$5.00, tiles: well you can spend lots but also get some very reasonably
priced ones on sale at $1.00 to $2.00 per square foot. Adhesive (but
thin-set mortar is better, and grout will cost under $30.00. Of course,
there are the hidden costs that you find when you remove the ceiling and
examine what is behind the problem.