If you want to grab everything via capture, you'll need to match everything:

while (<FILE>) {
    if (/(.*$find.*)/) {
        $header = $1;
} elsif (/^\s{38}\S*/) {
    print NEW "$header\t", $_;
    }
}
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Note that . matches every character but a newline, so there's no need to worry about chomping. See s in Modifiers in perlre.