For photography, the important factor is usually not whether it’s exactly new moon, but whether the moon is above the horizon.

A 1–2% crescent close to new moon is very near the sun in the sky, so it’s typically visible only briefly around sunrise or sunset. That means there are usually plenty of fully dark hours when the moon is below the horizon and contributes no moonlight at all.

The darkest conditions are when:

• the moon is below the horizon, and
• the sun is at least 18° below the horizon (astronomical night).

So in practice, the night before new moon, the new moon night, and the night after are often all effectively the same for dark-sky shooting if you choose a time when the moon is down. The difference between 1% and 2% illumination is usually far less important than simply checking moonrise/moonset and the length of astronomical darkness at your location.

Bottom line: yes, the adjacent nights should be fine if you schedule your shoot for when the moon is not above the horizon.