I’m also having this issue. I can log into my main on the Moogle server just fine, but my Adamantoise character continually gives me the “cannot connect to server” error, and when I was last playing, I was attempting to complete an instance for my class quest and DC’ed in the middle of it.

The character is Lucina Liera on the Adamantoise server, and I have no issues running the game on Moogle with no background apps open.