Can the moon shadow angle change?
Yes. The moon is lit by the sun, so as its position relative to the sun changes, so does the angle from which it is lit.

I recommend reading the Wikipedia page on lunar phase for a good overview, some data, and some good references. This diagram from that page more or less sums it up — the bottom row shows how the moon looks from the earth over the course of a lunar month:



Image is CC-BY-SA by Wikipedia user Orion 8.

The angle of the shadow depends on the time of the observation, and many lunar calendars seem to rotate the lunar images (as above). But this one gives a hint to the shifting relative sun position:



Image is of unknown origin, found here. Unclear rights; fair use.