Without looking further at your code, I think you will likely need to prepend the current path when checking for the file/directory type and when you call your subroutine recursively:

use File::Spec;

...
foreach my $FILE (@LIST)
    my $FULLFILE = File::Spec->catfile($FILEPATH, $FILE);
    if(-d $FULLFILE){                    
                    #Don't wanna have current directory and the one be
+low
                    next if $FILE =~ /^./;                    
                    #print("Current Directory: $FULLFILE\n");
                    push(@FILELIST,@{&readin_directory($FULLFILE)});
                }        
...
[download]

Is there a reason you're not using File::Find, which implements mostly what you've done?