The New York law simply codifies existing Roe v Wade abortion rights into state law. It's a hedge against the Supreme Court reversing Roe.
The Virginia bill, which has not been passed, would actually do nothing to change the time frame under which a fetus could be aborted. It was always technically legal in Virginia to have an abortion up until the day of birth. But the law in Virginia requires that late term abortions be approved by three doctors who testify that going through with the pregnancy would cause irreversible health consequences for the mother. The proposed law would reduce this to one doctor and would have reduced the standard of harm needed to approve a late term abortion.
On the general note of late term abortions, the anti-abortion activists push deliberately misleading information. First off, late term abortions are very rare. IIRC only something like 5% of all abortions are late term abortions, and while theoretically possible, fetuses aren't being aborted the day before birth. Second, anti-abortion activists make it out like late term abortions are just elective ones where irresponsible women wait until the last minute to decide they don't want a child. That's not true. Late term abortions are almost all cases where there's some sort of incredibly severe health defect with the fetus. For instance, a few years ago a Congresswoman talked about getting an abortion because it turned out that the child she was carrying was essentially going to be born without a brain. This is a child who they very much wanted, but this developmental abnormality meant that an abortion was really necessary. That's the reality of late term abortion. You don't go through six or seven months of pregnancy and then decide that you don't want the child. Instead it's that you find out that there's something wrong which will prevent a normal birth and necessitates an abortion