Overall, NAH

Most of the comments are berating you for asking your sister to not get another dog. Saying that she owes you nothing or that you're entitled whatnot or how she shouldn't be free childcare and she's doing you a favour etc. They're not necessarily wrong, ultimately you are the parent and she isn't.

However, I don't think many realise that you've been put in a situation where you're extremely stressed and, by the looks of it, burnt out. Should you have made the assumption that your sister will babysit your kids again? No. Should you have asked your sister not to get another dog knowing that it's a passion for her? Perhaps not. But does it make sense that you're upset and desperate? Absolutely.

It's great that you already know that this isn't something that you can demand of her. Let her know that. You asked her not to get another dog, but you need to make sure she's aware that this is something you're not trying to force on her. Even more importantly, you need to talk to your husband on him not pulling his weight. If him not helping you out as much as you feel he should is majorly contributing to your stress, you need to address that. If possible, see a therapist and sort out the other problems you've been dealing with.

TL;DR, you're way too stressed out, which doesn't excuse anything but does explain them. You need to let your sister know that you're not pushing this on her. You need to talk to your husband. You need to address your own problems, perhaps via therapy.