I recently installed a new vanity in my bathroom. The old sink sat a few inches higher and now the sink will not drain. Is the outlet for my drain too high? If so, do I need to cut it and lower it?

Compare the height at which your drain pipe comes out of the wall. The tailpiece from the sink can not terminate lower than that point. You can shorten the tail piece and attach your trap but the trap exhaust must be at the exact height of the pipe from the wall.