obs-browser: flush panel cookie store on exit

[zavitax]

If cookie store is not flushed, state might become inconsistent: reading from the cookies file will become impossible. No errors will be returned, yet no information will be available, and no information will be stored: the cookie manager will appear to be empty at all times and the Cookies file modified date will not update moving forward.

This will result, for example, in Twitch chat window re-opening for users which connected their Twitch account, but not having the auth-token cookie available: once the user will try to send a message to the chat, they will be redirected to Twitch login screen.