I am having the same problem too. The plugin says it’s connected but nothing is sent to the constant contact lists.
If I follow the link to “authenticate” the app provided in the comments from Constant Contact, it simply takes me to the download page in www.ads-software.com to download the app.
Which is already downloaded and supposedly connected. If I try the “connect store” from inside Constant Contact, again it takes me to the download page on www.ads-software.com. It’s a perpetual loop to the download page and nothing works. It makes me think mailchimp is worth a try over this hassle. It hasn’t worked for months and downloading and uploading every month is a hassle I don’t need.