in reply to Is there a better way.

Maybe:
chdir(Directory) or die "Directory: $!"; @uploads = glob("*$caseNumber*");
or
my @uploads = map { $_ =~ m!.*/(.*)! } glob("Directory/*$caseNumber*") +;
If your directory or $caseNumber have spaces in them then you will want bsd_glob, which you can get by:
use File::Glob qw(:glob);