Try republishing your site. Publishing to staging ( webflow.io ) alone should be ok.
I think of Webflow as a series of different modules that talk to each other using different channels. Some of them are one-directional, as in how the Designer knows about the CMS, and lets you use that content- but the CMS doesn’t know about the site design.
Other things like site settings rely on the published site view. If you haven’t published since your slug change, it may still see /mydesign-old as existing.
Also, you can check to make sure you don’t accidentally have any other pages with that slug, by publishing and trying to access your staged site, e.g. https://mysite.webflow-io/mydesign-old.