I don't understand in what way Mrs Clay depends, for the continuance of a much more comfortable situation than she was in formerly, entirely on Elizabeth and Sir Walter.
A more comfortable situation that she was in before her husband died? Do you mean that she is trying to get Sir Walter to marry her?
I can't find any reference in the text, however, that Lady Russell (or anybody else) ever thinks about Mrs Clay's two sons.
I might be a bit thick but I just don't grasp your meaning.