Their support says it is still possible to get the information we need for eNews to work correctly, but I was unable to find it and Constant Contact did not respond to requests for further information.
As of the last check I made, there was no way to embed a form to subscribe to a Constant Contact list from within a webpage—only have a link to a form hosted on a their service. With that type of change (and their lack of response for more info), I'm afraid it does seem that Constant Contact made a change in policy that makes it virtually impossible to use in our case.
As far as I've heard, forms already setup still work, so the change isn't technical, apparently.
It is possible that they have restored the ability to see the web form, but I haven't checked.