Can't login to Admin area, stuck in "Database update required" endless loop

My site was updated automatically to version 3.9.6, and now I can’t login to wp-admin. I got a “database update required” notification page when I tried to log in, with a clickable “update wordpress database” button. I clicked it, got a message that said “Update Complete Your WordPress database has been successfully updated!” with a “continue” button, but when I press that button, I am taken right back to the “database update required” screen. I have been through many rounds of this in Firefox and in Chrome, I’ve flushed the cache and tried again, but I can’t seem to log in. Any help would be appreciated!