Should I manually dispose the socket after closing it?

Calling Close internally calls Dispose so you don’t need to call both. From .NET Reflector:

public void Close()
{
    if (s_LoggingEnabled)
    {
        Logging.Enter(Logging.Sockets, this, "Close", (string) null);
    }
    ((IDisposable)this).Dispose();
    if (s_LoggingEnabled)
    {
        Logging.Exit(Logging.Sockets, this, "Close", (string) null);
    }
}

If possible you should use the using pattern so that you always call Dispose regardless of any exceptions that might occur.