Welcome to the community, Sian!
I’ve removed specific mentions of your previous email provider from your post, since mentions of competitors are prohibited by our Community Guidelines.
There isn’t a way to move subscribers between lists using Campaigns, but you can manage your subscribers in one list using tags and segments to determine which Campaigns and broadcast messages your subscribers receive. For instance, you could set up your form Campaign to remove subscribers when they receive a ‘quiz’ tag, and you can set up your quiz to tag subscribers when they fill it out.
Then, you could have a separate Campaign set up to trigger when a subscriber gets the ‘quiz’ tag so that they can receive different emails.
If you want the subscribers from both Campaigns to receive broadcast messages after they complete the Campaigns, then you can set your workflow up for this as well.
We do also have an ongoing beta for an update version of the Campaign editor that supports branching, which may help to set up this workflow all in one place. You can sign up for the beta here to try it out!