You are right,
Burlington will give you the best Redwood experience and the Visitors Center at the entrance is a must see.
There are some walk up sites that you will fit in,55,56,57 also 17 and 18 near the entrance.
Site# 17The larger 33 ft sites are on up the road, there is site 21 that is a pull thru, then 22,23,24,26,27,28,29,30,32,37,38. The rest are mostly for tenting. Do not go down the loop where 42 thru 47 are at.
The first time we were there we parked on the road and walked in and picked our site. The office was not open but there is always a list of the open camp sites and the dates they are open.
Albee Creek is in the open with some sites in the woods mostly suited for tent camping,it is interesting because of the meadow,it was an old apple orchard and in the evening the deer come in and forage.