Show your current list of cookies.
Clear the list of cookies.
Create a cookie function that accepts cookies that exist only during the current session and rejects the rest. Use Echo to report whether the cookies are accepted or rejected.
cookieFilter[cookie_Association] := If[cookie["ExpirationDate"] === Automatic, SetCookies[Echo[cookie, "Accepted"]], Echo[cookie, "Rejected: "] ]
URLRead["amazon.com", CookieFunction -> cookieFilter]
URLRead["user.wolfram.com", CookieFunction -> cookieFilter]
Now $Cookies only contains the cookies with no explicit expiration date.