To clear session (such as HttpOnly cookies), you can use InternetSetOption() from wininet.dll.
private const int INTERNET_OPTION_END_BROWSER_SESSION = 42;
[DllImport("wininet.dll", SetLastError = true)]
private static extern bool InternetSetOption(IntPtr hInternet, int dwOption, IntPtr lpBuffer, int lpdwBufferLength);
and use this method whenever need to clear session.
InternetSetOption(IntPtr.Zero, INTERNET_OPTION_END_BROWSER_SESSION, IntPtr.Zero, 0);
webBrowser1.Document.Window.Navigate(url);
Hope this will help you