Teacher moving with husband to Amman

Welcome on board 

I can't help you about teaching and setting up courses but I can give you some information about transportation and outdoor activities.

There are several threads about buying a second hand car so I suggest you browse through these. In the handy tools section, sub classified,  you can find offers of cars.
You can try that too. If you can't find what you're looking for, you can always come back and ask.

Outdoor activities:

I am not sure about softball and disc golf but you can definitely hiking here. Not only Petra and Wadi Rum but there are many wadis to explore. One of the most beautiful and famous is the Wadi Mujib where you hike trough water and behind a small waterfall.
Depending on your interests Jordan has a lot to offer in many different ways.