Google searches for mountain view CA apartment rents as an example of hyper dense urban life far from green areas seem to indicate $2K to $6K/month is quite reasonable. Similar google searches for rural Wisconsin cabins indicate a thousand bucks per week will get a cabin on a lake in the woods with a pontoon boat, wet bar, and hot tub. You don't have to worry about your schedule if your closest neighbor is a quarter mile away. Internet might or might not be a problem.
The problem with spending a couple months at some New Orleans bed and breakfasts is you'd be close enough to be reminded everything is shut down, and noise level. Might be easier to stick with rural retreats and adventures.
The united states lighthouse association website lists rentable lighthouses if you really want to get away from it all. Again, internet via satellite may or may not be a problem. The entire lighthouse on Charity Island in Michigan can be rented for not too much more than a Mountain View apartment.
You could do a very small house boat on a lake for $4K to $6K per month off season. There is no shortage of houseboat rentals and lakes and parks to visit. Could try the mississippi river...
In all fairness I did spend yesterday in a park shelter at a county park with excellent wireless working away in 70 degree breezes, but there are many days when it rains or its 100 or 0.