Yeah, it's a bit of a mess, the team are looking into fixes. In the meantime it appears a downgrade is a reasonable workaround. Links to known-good versions:
Beta? I upgraded from one of these normal popups that come when I start the app. When I do "Help" -> "Check for Updates" it still offers me 1.8.1 without any beta label. Am I overlooking something?
The team now acknowledged the problems on Twitter and is said to be working on fixes: https://twitter.com/sourcetree/status/699624992003244033
("Funny" that this is the first public mention of @sourctree of any update - nothing on the blog, Twitter feed...)