Dave and Phil have it right.
Our solution is to visit as few paid campgrounds as possible and stay as far from crowds as we can. Only going to town for gas, water, laundry and fresh veggies. No problem with a month or so on the road in the back country.
I always study google maps and Earth, watch YouTube videos if any exist for places on our general route and plan, plan, plan, which usually lasts the first couple of days if that long. As I drive, my wife carefully studies maps, paper and avenza on a gps capable samsung tablet. I have learned to be flexible and follow her lead when she spies something I did not notice when doing all that planning. We end up in some amazing places, such as a two-site USFS camp, with a clean pit toilet, that we had all to ourselves in the central Oregon forest last month.
Our near month in and around Death Valley this spring had us camping away from people pretty much all the time. Once some guy came by & camped about a quarter mile away. And we chose to spend a nite at Tecopa, where we met some interesting people... otherwise we may as well have been the only people overnighting in the park, by all appearances. We have had similar experiences in Nevada, Utah, Arizona, Oregon, Idaho and Montana.
I don't know how firmly attached you are to having a rental 'base camp'. We think of our pop-up as a portable cabin in the wilderness.
Good luck in your search.
Tony