So, I'm having similar issues. Not only can users not stay connected, but they also log in and it takes them back to the screen to do it again. Cookies are enabled in these browsers and it always works the second time.
I checked the config.inc.php file and all seems to be set properly (SESSION_PATH, FULL_PATH, FULL_URL, COOKIE_stuff). Other ideas? I'm using 7.2.2 at horizonsk8school.org.
Les