Welcome to the forum and congrats on your new Explorer.
How were the tires balanced? Just static balance or spin balance or speed balance? Just because one machine says it's balanced doesn't mean it is, it can be a good idea to double check on another machine. If you get anything other than a perfect 0.00 on a digital spin/speed balancer, it's not balanced.
You can get shakes from lots of other sources though, the universal joints on the driveshafts being one of them. If they are dry and rusted, they will cause a vibration at one certain speed like you're experiencing. The rear driveshaft only uses two u-joints, so replacing those isn't too expensive or difficult. I'd suggest getting Spicer U-joints. The driveshafts themselves are steel and often are rusted badly, so cleaning them with sandpaper to get off all the flaky, scaly rust usually puts them back in spec. Driveshafts are also balanced, and they should have circular weights on there for balance. If these weights fell off, the driveshaft might not be balanced and could be what's causing it. If the weights are fine, cleaning up the driveshaft (you can also paint it with some anti-rust paint to prevent future rust) might solve the problem.
You should also make sure there is no play in the slip joint, where the front of the driveshaft goes into the rear. It can also be a good idea to remove the protective boot and slide the driveshaft apart, cleaning the splines and using slip joint grease (Ford sells it just for driveshaft slip joints) on them. This solves a lot of noise and vibration issues as well.
You also want to check the driveshaft yokes at the transfer case and the rear differential. If they are leaking then the seals need to be replaced, if they have any more than a tiny tiny bit of wobble to them they need a new bearing.
You might also want to go ahead and replace all the fluids in the transmission, transfer case, and front/rear differentials. Synthetics aren't very expensive and can make a difference in noise and driveability. I'm not sure that old fluid would cause a vibration, but it's worth doing anyway, since most of the time, these fluids are never, ever changed for the entire life of the vehicle, and eventually they do cause strange noises, vibrations, and other issues.
