I've been testing this more, and found that when I send the newsletter to the administrator only, there is just one newsletter sent out.

When I send the newsletter to my test group each newsletter is sent out twice.

My email logs show two newsletters being sent out to each address, with a delay of about 5 seconds.

I've also tested the mail function sending a message to my test group. No duplicates there, so the problem is with the newsletter mod.