The answer to "what tools allow you to manage database migrations with CD" should not be "don't do database migrations with CD" or "roll your own toggling features."
I apologize for the disappointing answer, but continuously deploying in the sense of constantly pushing code to production multiple times a day isn't just a feature you add to your stack. It's not just another tool. It requires an engineering culture that is focused on it. We have testing, and automated tools for sure, but we have A LOT more tools for logging, and robustly handing errors. When a bad commit is pushed into production, I can pull up a graph that shows me the number of errors, and the full read out of every occurrence of it. I can also revert that code very quickly. We also have a team that does nothing but monitor the code, and the deployments. There are a bunch of tools you can find probably... but my point is that it's something we're thinking about from the design stage... because from design to production it might just be a few days. Major features will get pushed in stages, its a very different way of doing things (at least for me). Before I was here, deploying was something I did maybe once every few months. The code had to be absolutely perfect, because it'll be a while before I can get a chance to fix it. Here, I deploy while I drink my morning coffee.
If I was starting a new company I would without a doubt push towards doing frequent deployments all day long.
That said, I would would never push for it at my previous company. At this company it's something they've been doing since day 1. It's always been built into our process, culture, and hiring practices. At my previous company the "agile" process was just a shortened waterfall process. Though many of the engineers were very talented, and responsible... there was more than a handful that I would never trust to "self test" their code, or to monitor it as it deploys.
I definitely like that the more I think about it. We try to do deployments every couple months, but changes are so big and risky we end up pushing off deployments to the next release just because nobody wants to risk it.
Sorry but that seems like a pretty crap answer.
The answer to "what tools allow you to manage database migrations with CD" should not be "don't do database migrations with CD" or "roll your own toggling features."