ELI5 how it’s possible to ever see a full or new moon

Since full moon means the sun is shining at the moon head on from the earth’s perspective, surely that must mean that at the time of a full moon, the earth right in between sun and moon. …but then wouldn’t the earth cast a shadow onto the moon, resulting in a lunar eclipse? How can we ever see a perfectly full moon?

Likewise, at new moon, surely the moon has to be right in between sun and earth, otherwise we’d see at least a tiny sliver of a crescent lit up. But then wouldn’t a perfect new moon always result in a solar eclipse?