The new limit of 3 devices running Dropbox applications applies to the devices that can be connected to the Dropbox product itself, via official Dropbox applications, such as the official Dropbox desktop client or mobile applications.
If your computer is using Dropbox Folder as a syncing method for Moneydance then as you have to have Dropbox client installed this count towards the limit. However, if you use Dropbox Connection and don’t have Dropbox client installed this does not count towards the limit. This limit does not apply to third party apps that connect to Dropbox via the Dropbox API such as the Moneydance Mobile APP. Your mobile devices would only count towards the limit if you had Dropbox official app installed on your mobile devices for another reason.
Any replacement for Dropbox needs to support Windows, Mac, Linux, iOS and Android, most don’t. In its current form the free Moneydance app for mobile device needs an API provided by the cloud server service, equivalent to the API provided by Dropbox.