If you are this concerned, change your credentials.
So, yes, responsible websites can mitigate session cookies being leaked.
That said, I am not impressed by Cloudflare's transparency which in this case consists of downplaying things, blaming Google and Taviso and not really taking responsibility.
This leak is being downplayed by webmasters because it's so incredibly bad that there's no way of handling it. The credentials of practically any internet user could have been leaked. The only "safe" way to handle this is to give everyone in the US new credit cards and SSN's and to reset accounts and security questions for every user on a site with cloudflare
This issue is a drop in the bucket when it comes to the amount of sensitive data leaked.