They are targeting specific methods of contraception, which they claim are abortifacient. They have displayed zero signs of wanting to ban contraception in general, as opposed to just specific forms of it. By contrast, the law struck down in Griswold v. Connecticut was an across-the-board ban on all forms of contraception.

I think it is unlikely that Griswold will ever be overturned – simply because it is unlikely any state will enact a total contraception ban. To overturn a precedent, it is not enough that a majority of Supreme Court justices think it was wrongly decided, you also need a case – which requires a state to enact and seek to actively enforce a law violating the precedent. But almost nobody wants such a law now. 1965 (when Griswold was handed down), 1873 (when the law it overturned was enacted) was a very different world – when most Christians still believed that contraception was a sin. Nowadays, the vast majority of non-Catholic Christians (Protestants, Orthodox, etc) accept contraception as moral at least in some cases. And while the Catholic Church teaches artificial contraception is inherently sinful, it displays no signs of any interest in having the government make that sin illegal (unlike the situation with abortion). You might find some crazy Catholic integralist advocating it, but the Bishops and the Vatican aren't going to.

The bans on (allegedly) abortifacient contraceptive methods could be challenged, but I expect SCOTUS will uphold them – and the majority will probably just uphold them as state regulation of abortion, not overturning Griswold itself. However, I think a lot of groups are going to refuse to support challenges, on the grounds that they don't want to give this SCOTUS an opportunity to overturn any more precedents they support, rather save it for the day (whenever it may come) when there is a SCOTUS majority more amenable to their views.