It is true that Constant Contact stopped generating HTML form codes but they also made a form generator separated from their site.

You can download the form generator by clicking this and the instructions on how to install can be found here.

If you've followed the instructions correctly you should be able to access the form generator. There you can create forms like you would normally do and at the end it will generate custom HTML form code that you take to integrate with HC.

Note: just copy from <form> tag to </form> tag paste it on HC.