I stayed at Furnace Creek for 3 days at the end of October without reservations - asked for 3 days & there were only 3 sites available (all dry), and only one would fit your trailer. Many more were available for 1 night stays. My favorite Death Valley campground Texas Springs (No Generators) was closed as was Sunset. I was told they may open later in the year - one ranger said they were closed because of poor scheduling; not enough maintenance people, another said it was due to budget.
Since the park gets more popular as it cools off, I'd check to see what is available.