$new::variable is a package variable belonging to the 'new' package (bad name btw). $another_variable is a lexical variable that you need to declare using 'my' because you are using strictures (use strict; use warnings; - highly recommended btw). The fix is just:

use strict;
use warnings;

use MyPackage;

$MyPackage::variable = 1;

my $another_variable = 2;
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Note that if package MyPackage uses strict it will need to declare $variable using our:

use MyPackage;

use strict;
use warnings;

our $variable = 1;
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