I am having the same problem with the WP Super Cache Plugin installed correctly. My mobile theme shows up on desktop sometimes and the desktop version shows on mobile (scaled).
My site: https://bit.ly/37vDTeq
Is there a way to work around this?

Settings op WP Super Cache
Caching Enable Caching
Cache Delivery Method Simple (Recommended)
Expert
Expert caching requires changes to important server files and may require manual intervention if enabled.
Miscellaneous Cache Restrictions
Enable caching for all visitors.
Disable caching for visitors who have a cookie set in their browser.
Disable caching for logged in visitors. (Recommended)

Don’t cache pages with GET parameters. (?x=y at the end of a url)
Compress pages so they’re served more quickly to visitors. (Recommended)
Compression is disabled by default because some hosts have problems with compressed files. Switching it on and off clears the cache.
Cache rebuild. Serve a supercache file to anonymous users while a new file is being generated. (Recommended)
Cache HTTP headers with page content.
304 Browser caching. Improves site performance by checking if the page has changed since the browser last requested it. (Recommended)
304 support is disabled by default because some hosts have had problems with the headers used in the past.
Make known users anonymous so they’re served supercached static files.
Advanced Enable dynamic caching. (See FAQ or wp-super-cache/plugins/dynamic-cache-test.php for example code.)
Mobile device support. (External plugin or theme required. See the FAQ for further details.)
Mobile Browsers

Perfect, thx.