Is there some way to carry over followers between addresses that I'm unaware of?

Yes: https://docs.joinmastodon.org/user/moving/

How exactly do you do this after the instance admin has banned you?

You don't. If your admin is a total dick and prevents you from migrating, you're screwed. If you have hundreds of followers, better reevaluate your initial choice of instance. If you have thousands of followers, host your own, or pay somebody to host it for you. This however doesn't matter if you're just getting started.

There are talks about detaching user identity from the instance. If this gets traction, this problem might be solved once and for all.