Can you anchor your boat to not Moore it?

No, you are talking about two different things. A moooring is a mushroom shaped anchor that stays in position even with tides changing direction of the boat. An anchor won't always do that and thus needs to be monitored and possibly reset.

In other Words, morrings are permanant, anchors are temp.

Also, you typically need permission to set a mooring from the town/marina etc...

In our town a mooring means tieing up at a dock in the community marina or at the yacht club. Anchoring is not " parking " at a dock site.