I believe that using a user-agent switcher should disassociate your browser, at least within reddit's automated "these users are the same person" system. You can get a chrome extension to do that. I don't believe reddit is doing extensive browser-fingerprinting beyond this as they would face considerable backlash by privacy nerds on the site for doing so.

Obviously don't use the old account while using the account that uses a spoofed user-agent.

Now, this might not stand up to a manual review if they have other factors they track too, but my experience has been that it does get passed their automated system which is likely what people are losing subreddits for. Admins aren't furiously stalking chapochat every second of the day to ban subreddits, it's an automated ban triggered with a timer on it to confuse people and make it harder to figure out why/what triggered it.