You can't move your content to Medium without affecting SEO.
- The domain ranking signals will be very different. You used to have your own domain with its own set a ranking signals. Medium.com has its own set of signals that you won't have much of any control over.
- Your content will be mixed in with everybody else's content.
- The links on your page will be very different. While you used to have full control over the links on your page, Medium will have that control if you move there. Instead of your blog pages linking to each other and your blog home page, on medium some of the links will point to other content such as the Medium home page and other articles written by others.
If you do move your content, you should redirect each page to the corresponding page on Medium to preserve your rankings as much as possible. This may not be easy to do. Your URLs on Medium will likely have a very different format than what you had on your own site. A simple domain redirect is unlikely to suffice. You say you don't want to run your own server, but in this case you might have to just to be able to issue all the needed redirects.
It is hard to say whether or not your articles will get more or less search engine traffic when hosted on Medium. I'd say that in general Medium has a decent search engine reputation, although it does host a lot of duplicate content.
When you blog on medium, you will be giving as much value to Medium itself as you will be getting for you. You may find it difficult to move off Medium in the future back to your own domain. I'm not sure what types of redirects they offer, but even if you can redirect, splitting content off a domain is never as straightforward as other types of content moves.
If you are serious about SEO, you should continue to use your own domain name. I'd suggest looking for a managed WordPress host in the range of $30 to $60 per year.