The White Mountain Park does allow backcountry camping, their guide on rules and regulations can be found here.
Yes, the huts are expensive, and usually filled with causal hikers and families. They do provide dinner and breakfast as part of the cost however. Many of them are basically mountain top resorts sans vehicular access. They do treat Appalachian Trail hikers with “free” accommodation and food however, which is nice.
I believe the shelters and tent sites only cost money when they have a care taker, and the costs are usually <$10US. This is generally in peak season (so when you will be there), to cover the costs of maintaining the grounds (and in theory, to limit use?).
Other than that no permits are required to hike in the area that I know of.
I’ve spoken to many caretakers and park rangers about backcountry camping and they are extremely reasonable about it. There are definitely many “backcountry sites” that have developed over the years, and they prefer you to use them if you see one. They are unofficially taken care of by the park rangers as well as those of us who use them and care about minimizing the backcountry camping impact. One ranger told me that he and some others definitely visit them from time to time to make sure they are kept in shape.
However, AMC hut employees are basically clueless, and many times spout off on it and are completely wrong, or just ill informed. They are generally college kids, and they probably mean well, but I would not take anything they say about things beyond hut life as fact.
If you do happen to see a park ranger on your hike, I highly recommend taking advantage to chat them up, they’re a wealth of knowledge and always seem to have some great stories about their experiences working in the Parks. Of course, asking them about the maintained backcountry sites in the area is also useful.