Is this what you meant?

if($usermail ne "") {
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This is going to send email when $usermail is not blank. From the stuff inside the condition, it appears that you're sending to every address entered so far. So, for example:

1. You enter u1@example.com, mail is sent to:
   • u1@example.com
2. Then you enter u2@example.com, mail is sent to:
   • u1@example.com
   • u2@example.com
3. Then you enter u3@example.com, mail is sent to:
   • u1@example.com
   • u2@example.com
   • u3@example.com

Then again, you say that the mail is not sent at all, so perhaps I'm way off in the weeds here.

I'd suggest that you use Data::Dumper to show the contents of @addresses before the attempt at delivery.

use Data::Dumper;

print Dumper \@addresses;
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It's a good idea to show $usermail before you push it into the list too.

warn "\$usermail = '$usermail'\n";
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Those two should give you a better idea of what's going on.