We were in Cooktown in 2012, stayed at the Big 4 Cooktown Holiday Park
http://www.cooktownholidaypark.com.au/ Not a cheap park, but one of the nicest we have stayed at, the managers at the time were lovely people.
If you want to stay on the bitumen with your van you really can't go much past there. We did a number of day trips without the van to places a bit north like Isabella Falls, and south east along part of the Bloomfield track. We really liked Cooktown, a couple of great museums and the view from Grassy Hill lookout shouldn't be missed.
We went to Lawn Hill in 2013, stayed at Adels Grove. The road to there via Gregory Downs, while gravel, was ok for on road vans as long as you took your time. The road between Adels Grove and Lawn Hill at that time was much much more corrugated, and we were very glad we hadn't tried to take the van into the National Park camping area. At Adels Grove we stayed in the Grove section, unpowered, no generators and shady down by the river, no set sites. We were a fair walk from the very good amenities block but we intentionally chose a secluded spot. The only water to fill tanks was up near the amenities. There are also set sites which allow generators and pets up above the Grove.
At Lawn Hill make sure you hire a canoe one morning to do the easy paddle down the gorge as far as the waterfalls. From there you can portage the canoe over a land section and paddle a further section, but we decided the canoe was too heavy for us to do that. Instead we had a swim and then ate our morning tea on the dock, with archer fish shooting at our bickies as we ate. The next morning we did the walk along the gorge, which gives you a different perspective on it than canoeing, both are well worth it. We also did a couple of the other walks in the park. Was without doubt one of our highlights of that trip.
