Very nice solution! Your interpretation of quoting is correct (I blundered before). Your solution does not work, as is, however, because $1 gets clobbered. The following minor changed version of your solution does work for me (note also the extra line of test data with your quoting example).

my $str = <<'EOS';
abc "a\"bc$xyz$abc$x$y" $bla'h, $blah
hello k\$nob
this is "\"'abc$xy$z\"" test "$"
a$$bc "$$abc$xyz$abc$" blah, $$, blah
abc "a\"bc$xy\\z$abc$x$y\\" $bla'h, "$baz"
EOS
print "$str\n\n";

    $str =~ s[
        \G            # Don't skip anything
        (             # $1 is unquoted text
          (?:
            [^\\"]+   # Uninteresting stuff
          | \\.       # Escaped character
          )*
        )(            # $2 is quoted text
          "           # Opening quote 
          (?:
            [^\\"]+   # Unintersting stuff
          | \\.       # Escaped character
          )*
          ("?)        # $3 is closing quote
        )
    ][
        die "Unclosed quote ($2)"   unless  $3;
        my $t = $1;   # changed line
        my $q = $2;
        $q =~ s[\$][\\\$]g;
        $t . $q;      # changed line
    ]xseg;
print $str;
[download]

which prints:

abc "a\"bc$xyz$abc$x$y" $bla'h, $blah
hello k\$nob
this is "\"'abc$xy$z\"" test "$"
a$$bc "$$abc$xyz$abc$" blah, $$, blah
abc "a\"bc$xy\\z$abc$x$y\\" $bla'h, "$baz"


abc "a\"bc\$xyz\$abc\$x\$y" $bla'h, $blah
hello k\$nob
this is "\"'abc\$xy\$z\"" test "\$"
a$$bc "\$\$abc\$xyz\$abc\$" blah, $$, blah
abc "a\"bc\$xy\\z\$abc\$x\$y\\" $bla'h, "\$baz"
[download]

as does my original attempt; however, the last line of this new test data trips up the earlier one liner.