Craigslist is your first and best source. Besides cruising the various categories of available housing, post your own ad in housing wanted. Be specific and sell yourself. Mention that you are a travel nurse. Travel nurses are highly desirable tenants or housemates. Not only do we have a license to protect, everyone knows we can afford to pay and will pay. You will find places by posting your own ad that are not advertised. Some landlords will cruise the housing wanted ads and pick good candidates to contact.
AirB&B, VRBO, and other such services have been used successfully by other travelers. It is a different business model and tends towards expensive very short term rentals. It is probably better to use those services in less desirable areas with lower demand. The way to approach it is to book a couple nights at a likely looking place. Now you can talk to the owner directly and potentially negotiate directly a much better price for a long term stay. I have done this a couple of times but never for longer than two weeks. But even traveling abroad, I've booked a couple nights and then switched to paying the owner directly for a better rate. There is an aggregator of such services called AllTheRooms that list on one page all the available housing to shorten this kind of search.
Personally, if Craigslist fails, I usually search for local property managers. Most of the time I am already local to the assignment and can pick up a rental list drive or ride around to look at the property and then arrange access. This kind of rental is a bit of a hassle as you have to pay for a credit check, and if you take the property, that address will now appear on your background check as a prior address. Since we are short term rentals, and most property under management is long term, the property manager has to call the owner to see if they are willing. Often the answer is yes and away you go! I have had a couple of great rental properties obtained this way. But I prefer dealing with private owners. Often they will waive a security deposit if I say I'm willing to pay in advance - thus reducing their potential hassle of collecting rent monthly, and more importantly to me, reduce the chance of no return of the security deposit after I've left the area and have little recourse.
You said you were not interested in corporate housing, but for the sake of completeness, there are a number of companies now in the business of supplying short term corporate housing to agencies such as Furnished Finders, Travelers Haven, and Med Temp Housing. They will rent directly to travelers, and even here, you will usually be better off than the agency providing the same exact housing because of the tax benefits of taking the housing stipend and keeping any excess stipend to use as you like.
Getting your own housing is certainly a hassle, but a rewarding one. Some travelers enjoy the luxury of traveling by RV, but that comes with its own set of hassles and generally will not save you any money over more ordinary housing.
Just to mention here a big caveat, tax free stipends only apply to travelers with legitimate tax homes. That should be a bigger concern to you than housing. Read more about tax homes on sites like PanTravelers and Traveltax as part of your planning prior to starting travel.