I’m not sure how Chrome handles cookies, but as long as the different sessions are using different sets of cookies you should be fine.
On Firefox, for example, the Multifox extension doesn’t work right because the profiles share cookies. However, I have Firefox (personal tumblr) and Firefox Developer Edition (xkit tumblr) open simultaneously without issue.
It should be pretty obvious if it’s not working—usually one blog will be unable to use One Click Postage if you’re having cookie issues. What do you mean by everything disappearing?