But, as you say, this won't give you the headers, so if you need the cookie - you're fucked. Add to that the prevalence of javascript and you might also be missing out on other headers/modifications that are happening that are not reflected in the html document.
I've noticed cookies are usually in the header under the "Set-Cookie" header...
The only other way I know that I could get that info, would be to actually go into the directory that the cookies are stored in, find it, open it up...
I'm curious why notreal said to not use Burp Suite. It seems to me like the route to go.