Put it in 2WD (drive it around the block to make sure only 2WD is engaged), and put it in neutral if you go with the approach of having 2 wheels on the ground. You'll need to pop off that shift lock cover and press the button to shift to neutral with the engine off.
I've moved cars across states like that and have always gone the flatbed/full trailer route. Those 2 wheel carriers always made me really nervous, especially with greater distances. This is what I used last time:
Either way, make damn sure the wheels are torqued down good and check on it every few hours for good measure. All of the road vibration and such will make it hop around a bit and wiggle its way loose, like with any strapped down load.
Maybe deflating the tires that are strapped down a bit would help...