>>Using 'cookieless=true' doesn't prevent you from using cookies - just means that the sessionId is in the URL not in a cookie. So if, for example, you wanted to store the user name per session you could use a cookie with the SessionId as the key and 'UserName' as a sub-key. Something like this.
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>So, we are talking about the ASP.NET session and not the Member.Session field I am using.

I don't know what you mean by Member.Session (?) but yes, ASP.NET session.

> Does this mean you see visually in each URL request a longer URL with the ASP.NET session?

Yes. If you allow sessionId to be stored in a cookie you will get the same id for new instances (or new tabs) for the relevant browser.
If 'cookieless=true' the session id is included in the URL and you get a different session for each instance or tab.

Don't know any other way in which you can maintain separate sessions for different instances.