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use strict;
use warnings;
...
    keys %{{map {$_ => 1}($str=~m/("[^"]*\$[^"]*")/g)}};

print Dumper(\@uniq);

$VAR1 = [
          '"c$d e$f$"',
          '"$a $a $a"'
        ];

...
my $str = '"$a $a $a" ab bc "c$d \"e$f$" c$d "$a $a $a"';
...
    keys %{{map {$_ => 1}
      ($str=~m/("(?:\\"|.)*?\$(?:\\"|.)*?")/g)}};
...

$VAR1 = [
          '"c$d \\"e$f$"',
          '"$a $a $a"'
        ];