Hello,
I also have the same problem. I have the latest ver of Evernote (desktop ver) for win 10 but after I login fine, it would start to sync automatically & then keeps asking for passwd multiple times. The strange thing is I do not have any issues logging into the web ver & android ver. The sync problem is only with the WINDOWS 10 desktop ver.
I have even removed & re-installed EV but still no luck. Any ideas how to fix it ? Pls see the attached screen shots.