You could always BCC yourself, or a monitoring account, on every e-mail sent.
Madmimi will send unlimited emails to 100 addresses for free.
Mailchimp free supports 2,000 subscribers and 12,000 emails per month.
Mailchimp has an API that can be setup to talk to Filemaker.
Both track bounces, recieved and opened emails.
http://icontact.com is another email service.
2012 Best Email Marketing Service Comparisons and Reviews