I find the north side much more beautiful. I've rarely had any problems with rain in November or other winter months (been there four times).
For me Kauai's "winter" means I sometimes put on long pants or a long-sleeved shirt.

If I had only a week or less on Kauai, I would stay on the Eastside, because that way you have access to both north and south sides of the island.

Of course, if you want "guarantees" of sunny weather, you are better off south or est.