Using an Xperia 10 II and the latest Sailfish OS (4.4.0) with the ESR78 version of the browser I’m able to log in to DropBox. I can both log in to use DropBox through the browser, as well as being able to add a DropBox account via the Settings app.
If anyone is able to test this again on the latest Sailfish OS 4.4.0 Vanha Rauma, it would be useful to know the results, since this should now be fixed.
@Steve_Everett has posted a bug report about this issue, so if your findings are relevant to the bug, then I’d be grateful if you could also post info to that topic.