I have found a solution that consistently works to solve this problem (on Windows). Posting it here to help anyone who is experiencing similar.

Before playing the game, open command prompt. You can do this by typing "cmd" in the search bar on your taskbar in Windows.

In the command prompt, type the following, then push enter:
ipconfig /flushDNS
The command prompt will then say
Windows IP Configuration
Successfully flushed the DNS Resolver Cache.
You can now close the command prompt and start Final Fantasy as normal.

If, after doing this, the problem persists, restart your computer, follow the above steps to flush your DNS cache when it boots back up, then run Final Fantasy before opening any other online applications (web browser, discord, etc). Once you are in queue to connect with a character, you can tab out and open up that other stuff that you want and use your computer normally.

I don't know if this will work for everyone, but this has worked very consistently for me. Flushing the DNS cache before starting the game has basically eliminated these disconnects, whereas I was getting them every hour or two before I started doing this. If I get DC-ed, a restart and a re-flush of the DNS cache prevents further DCs until my next play session very consistently.

Hopefully this helps someone else out there!