I just pushed an update for the legacy branch of Postman (v0.8.3) on the Chrome Web Store. The update has a bulk export option which combines all your Postman data in one file. If you are upgrading to the newer version of Postman you can now import this file. by going into the "Settings > Data" screen.
One workaround is to use the Postman Proxy to set the cookie header with the cookie value obtained from the browser.
Some users have reported the unavailability of the Network Inspector. Unfortunately, Chrome disables Chrome DevTools by default for installed apps. Type chrome://flags in Chrome, look for "Enable debugging for packed apps" and hit enable. DevTools should now be available inside Postman.
Stay tuned for updates and if you encounter any issue do report it on Github or Twitter! Replying to reviews is not possible inside the Web Store and hence I would not be able to inform you if there is a fix available for your problem.