With your experience, and some ingenuity, I doubt you'd have much problem finding places to free camp in the US. The rule of thumb is the same here as there: be as inconspicuous as possible, and leave no trace. As English speaking 'foreigners,' Brits I assume, I dare say your reception here may be more generous than for natives, people more willing to go out of their way a bit to help.
One tactic I've used to avoid camping fees at public parks, and still avail myself of needed services, is to just free camp somewhere nearby, and buy a day pass, usually $5 or less.
Small town city parks are often hospitable. I check them out to get a 'feel' for their potential, especially for concealment. Adventure Cycling Association maps are great for this sort of information, as well as much, much more. Highly recommended for non-natives crossing the US on a bicycle.
Www.warmshowers.org can be a great way to meet people and be hosted overnight, with either a tent spot or warm bed. Plenty of advance warning suggested.