First off, they seem to want to do nothing at all about the short-sighted buffoon jackass son who seems Hell-bent on making her life miserable because he's short-sighted and stupid enough to think that Lizzie is actively trying to destroy him utterly merely because he's asked to share for once. According to those two zombies, children fight for no real reason and become friends when they grow up so nothing needs to be done to address the issue.
Second, we have to deal with the fact that despite her refusal to admit to it, Lizzie can be awfully God-damned possessive when she wants to be. Most of the conflict she's going to have with Dawn over the next decade or so stems from the fact that she simply doesn't want to share her friend with other people because she thinks that since Dawn hangs out with someone new, she must actually like that person best. This sort of genetic inability to understand social norms leads us to the last sort of problem.
Said problem is that she shares being extremely short-sighted with her older brother. It took about a month's worth of strips for her to actually notice the fact that Anthony had this huge, sloppy and embarrassing crush on her because she, despite her claim to be aware of her surroundings, simply was incapable of noticing the obvious when it's staring her in the face.
The reason for this refusal to actually notice her surroundings seems, I should think, to be explained by her habit of only listening to people who actively want to make her feel bad about herself. Just as Phil can never convince Elly that her childhood was not one of servitude, misery and misogyny and John can never convince her that she's not disgustingly fat and ugly and unworthy of love, no one can convince Liz that she's not ugly and stupid and worthless. You simply cannot argue away depression any more than you can install a sense of right and wrong in a delicate genius. You might as well try to convince Connie not to envy Elly as bitterly as she does.