I have a new Mac Mini with Catalina and I can't open my Dropbox app. I can put files into it in my Finder window but they are not shared to my other devices. The icon in the upper menu of my screen is gray.
I have tried using the solutions suggested. I have installed the latest version of Dropbox and I tried opening Terminal and pasting the code sudo mv ~/.dropbox ~/.Trash/dropbox.old and entered my admin password and that was supposed to then create a prompt that would allow me to open the Dropbox app in Applications but it would not open.
Just another crazy day in computer life. I started the article you referred me too and the instruction was to sign out first. (I had previously done that and then re-launched but was still getting the No permission to sync message.) This time however as I clicked on the grayed out icon it gave me the opportunity to sign in. So that's what I did and then got a message saying that I was signed into too many devices. It turned out two of them were the same, my new Mac Mini. I removed one and then, voila, everything is working just like old times!