Dawn,
Some mobile home parks have RV spots which would probably be vacant during the winter months. Some parks have a rather "seedy" reputation but most are well kept. I'd check with the local police department first, then choose based on their recommendations. They might even know of any individuals with room for your rig.
Other possibilities might be discovered by checking with the churches in the area.
Checking with real estate offices might give you a lead on a property which a previous owner had set up for an RV but has been
sold to someone who is not using the spot.
Another idea just occurred to me. If you were so inclined, with the 2 dogs you might provide "security" at a construction site. Usually you would be parked behind a chain-link fence with an outhouse, water and electricity. A cell phone for communication would be a plus.
Wishing you success,
Kurt & Ann K.