Check all your permissions on the server and also check the PHP Session settings on the server. If you can't hold a session, you may have potential problems holding a login.

Can you login as a CUSTOMER (from the catalog side). If so, then the previous concept might not bee worth looking into, and you should look exclusively within the admin files and configuration themselves.

Make sure your PHP setting have the temp path set (to hold session data etc..). I found one time with another site that the temp path was not set correctly on a server, and as a result, the login would perform similar to what you are describing.

Mind you, that wasn't a Zen Cart site, but it may be worth looking into.

Testing login between customers and admins will at least give you a place to start narrowing down your issue.