The only place you need to setup the List ID from Sendy is in the Sendy - System plugin. In there, make sure you've given it a 'pretty' name "My List #1" and the full (about 20 character) List Id. Then, in the Sendy Module you need to select the "My List #1" option.
If you're sure those fields are setup properly in the system plugin, then we'd likely need to investigate further what could be going wrong. Iin that case, can you Private Message us admin credentials for your site? You can PM me by clicking the mail icon under my picture to the left.
I'm sorry for the delayed response. Thanks so much for access to your admin area. We found the issue and updated your release of our Sendy integration with a minor point release we'll make available soon. The configuration tabs in the module and plugins should work now and you should be able to configure things as expected.
Hopefully this gets you going, but if you run into any other issues, just let us know.