Check out the cabins at Julia Pfeiffer Burns State Park. This is a beautiful campground that truly captures the spirit of Big Sur, as you state you are looking for. If you are not going during peak season you will hopefully be able to reserve something. Also remember that Big Sur is only a 20 minute drive from Carmel.....
There are also several private campground/cabins in the area but they aren't as nice as those at the state park.
We just returned from our 15th annual camping trip to the Riverside Campground in Big sur and the cost was high ($45/night) for small sites and filthy unkempt bathrooms, where 2 of the 3 showers didn't work. Be careful. This campground used to be very well maintained but evidently a change in management over the past few years has changed that. We attend with 5 families from across the state so we are willing to put up with it as we can reserve specific sites together.
LA to Big Sur will take 5-6 hours, depending on traffic. the last stretch on beautiful winding Highway 1 could be slow as it is only a 2-lane road and slow vehicles don't always pull aside to let others pass.