You haven't really told us what you want to do with the files. "Separate"... does that mean put them into different directories? Hold their filenames in different arrays? Break the ones with multiple headers into multiple files? We only know what you tell us.

Once you've opened the directory and gotten a list of files, you could use this snippet to proceed.

FILE_LIST: foreach my $filename ( @files ) {
    open FILE, "<", $filename or die "Can't open $filename. $!\n";
    my $headers = 0;
    while ( my $line = <FILE> ) {
        $headers++ if $line =~ /\bFULL\b/;
        if ( $headers > 1 ) {
            # do whatever it is you intend to do with
            # multi-header'ed files.
            next FILE_LIST;
        }
    }
    # Do whatever it is you intended to do with
    # single-header files.
} continue {
    close FILE;
}
# Now you're done.
[download]

As you can see, Regular Expressions! are only a very small part of making this thing work for you.