Hooray, I figured it out!
Hopefully this helps someone else out there one day:

I had to first:

my $thisItem;  #<-- this is imperative!

then:

$thisItem = $folder->Items($i);
if ($thisItem->{'Unread'}) {
...etc...
[download]

and replace every instance of

$folder->Items($i)

with

$thisItem

Then it works like a charm!!
Also don't forget to save changes at the end of changing each email:

$thisItem->Save();

Yay!