Here's a partial workaround, but you will have to do it every time before you start Galaxy until they fix the underlying bug. Every time you close Galaxy and restart it, the integrations will break again.
Close Galaxy and open this folder:
C:\ProgramData\GOG.com\Galaxy\storage\plugins
Delete the .db files for the integrations that won't connect.
You can then start Galaxy and they won't be stuck spinning. You will have to re-connect and log in again, but your settings should be saved, since you are only deleting your saved credentials.
The ProgramData folder is hidden by default, but you can just copy and paste the path into Run (Win+R or right-click start button->Run) or the File Explorer path bar.
If anyone is curious, you can use a tool like DB Browser for SQLite and open the .db files. They are regular sqlite databases. The ones the plugins directory contain only one entry, and that is a JSON formatted credentials strings that contains your token, your login id, maybe a couple of other fields -- it's a little different from each service.