I have the same problem as this post from a few years back. Basically, my strategy is that since I'll only be livestreaming a couple hours a week, I would have one EC2 instance just for Wowza that's up only when I'm actively streaming, and a separate micro instance for my 24/7 website. When I want to bring up the wowza instance, I can make a trivial edit to the website to point to the now-running Wowza site.
Problem is, the pre-built EC2 instances for Wowza cannot be stopped, they can only be terminated, meaning any changes I've made to them (like configuring Wowza, or styling the livestream page) are lost forever. It also means I can't hold onto an elastic IP address, since I wouldn't have an instance to associate it with (and I'd be charged for not using the IP).
I'm new at EC2, so I'm sorry if this is a stupid question that anyone who uses EC2 should be able to figure out. But it seems like short-lived Wowza instances would be such a common need, the situation can't be as bad as it looks, right?