The ON seems to indicate that Elizabeth will not marry.

"It would be well for the eldest sister if she were equally satisfied with her situation, for a change is not very probable there. She had soon the mortification of seeing Mr. Elliot withdraw, and no one of proper condition has since presented himself to raise even the unfounded hopes which sunk with him."

And this is all we hear--the rest can be conjectured. I suppose Elizabeth COULD live by herself, if she wanted, couldn't she? Rather like Lady Russell? She might be contented staying in Bath, getting another Mrs. Clay-type person to "flatter and follow" her and generally be a companion. I can't picture her going to live permanently with one of her sisters. If she did that, she'd perpetually live in a condition when she does not have precedence. So the other two options are living by herself, or constantly traveling between the houses of her friends and family, like Edward Ferrars and Mary Crawford did. That is the scenario I personally think most likely after the death of Sir Walter. I see her spending the bulk of her time with people such as Miss Cartaret or other distant rich relatives in London and Bath, and then spending the occasional month with Anne or Mary in between other engagements.