First, you can have two passports; not just a tourist and official. When we lived in Belgium, we needed to leave our passports two weeks at Russian Embassy for visas. As we were traveling to Poland in the interim, I was able to request additional passports for the two of us. They cost $85 each and are good for only two years so I don't recommend unless necessary.
As for travelling using both; it is probably a good way to end up in jail especially when USA joins the rest of the world in recording departures.
For Israel, if you want, they will stamp on separate paper. I don't worry about Israeli stamps in my passport as I have no plans (currently) to go to countries where it has an impact. BTW, you can go back and forth to Jordan overland with no issues.