Is there a way to track two domains in Google Analytics without passing the cookie information in the URL?
Since GA doesn't support 3rd party cookies and the cookie passing isn't what you want (it doesn't work in all the cases anyway), you're left with either changing the tool or doing some highly customized stuff, e.g. something like this:
Domains 1 and 2 share a database (table) where the GA-cookies of each visitor are kept. If the visitor goes from domain 1 to domain 2 (or vice versa) you check (the matching of visitor should be done with IP + User Agent + maybe referrer) for cookies in DB at the server side (before GA has time to execute) and write them if it matches, so that GA will have the same cookies available.
Remember that the cookies contain information, like the number of page views during current visit, which would have to be grown before writing the cookie.
You have to do this every time the page loads and the referrer is not internal.
It would work if you really want to do this, but save yourself some trouble and headache and get a proper WA system.