Unless I'm mistaken, this snippet should go a good ways toward solving your problem.

use File::Spec::Functions;
use File::Copy;

our @files = glob "*.txt";

mkdir for qw( multiple single );

for my $file ( @files ) {
    if ( 1 > number_of_headers( $file ) ) {
        move( $file, catfile( "multiple", $file ) );
    } else {
        move( $file, catfile( "single", $file ) );
    }
}

sub number_of_headers {
    my $file = shift;
    my $count = () = slurp( $file ) =~ /FULL/g;
    return $count;
}

sub slurp {
    my $file = shift;
    local $/;
    local *SLURP;
    open SLURP, "<", $file or die "Couldn't open $file for reading: $!
+";
    my $content = <SLURP>;
    close SLURP or warn "Couldn't close $file: $!";
    return $content;
}
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