You can use the following pattern to slurp and spew a file:

use warnings;
use strict;

my @files = <Test*_Copy>;
for my $modfile (@files) {
    # read
    open my $ifh, '<', $modfile or die "$modfile: $!";
    my $data = do { local $/; <$ifh> }; # slurp
    close $ifh;
    # modify
    $data =~ s/Marry Had A\nLittle Lamb\nShe Was GOOD\n/CHANGED!!!\n/s
+mg;
    # write
    open my $ofh, '>', $modfile or die "$modfile: $!";
    print $ofh $data;
    close $ofh;
}
[download]

(Slurping can also be shortened to my $data = do { open my $fh, '<', $modfile or die $!; local $/; <$fh> };, though that's a bit less readable if you're not used to it).

Or you can use a nice module like Path::Class or Path::Tiny:

use warnings;
use strict;
use Path::Tiny;

my @files = path(".")->children(qr/^Test.*_Copy$/);
for my $modfile (@files) {
    $modfile->edit(sub {
        s/Marry Had A\nLittle Lamb\nShe Was GOOD\n/CHANGED!!!\n/smg;
    });
}
[download]

Or you can use a oneliner (see also $^I):

perl -wMstrict -0777 -i -pe 's/Marry Had A\nLittle Lamb\nShe Was GOOD\
+n/CHANGED!!!\n/smg' Test*_Copy
[download]

Or you can use my module File::Replace for atomic updates when the filesystem supports it:

use warnings;
use strict;
use File::Replace 'replace3';

my @files = <Test*_Copy>;
for my $modfile (@files) {
    my ($infh,$outfh,$repl) = replace3($modfile);
    my $data = do { local $/; <$infh> };
    $data =~ s/Marry Had A\nLittle Lamb\nShe Was GOOD\n/CHANGED!!!\n/s
+mg;
    print $outfh $data;
    $repl->finish;
}
[download]

Or you can open the file in read-write mode, truncate it after reading, and then write it - though note I see this being done very rarely. (It's useful if you want to flock the file while editing. That can be as simple as adding use Fcntl qw/:flock/; at the top of the code and flock($fh, LOCK_EX) or die "flock $modfile: $!"; immediately after the open. Keep in mind that flock locks are merely advisory, though!)

use warnings;
use strict;

my @files = <Test*_Copy>;
for my $modfile (@files) {
    open my $fh, '+<', $modfile or die "$modfile: $!";
    my $data = do { local $/; <$fh> };
    $data =~ s/Marry Had A\nLittle Lamb\nShe Was GOOD\n/CHANGED!!!\n/s
+mg;
    seek $fh, 0, 0 or die $!;
    truncate $fh, 0 or die $!;
    print $fh $data;
    close $fh;
}
[download]

Note that glob has several caveats, though as shown in this code it's ok.

Updates: Added info on flock and $^I.