How about the other OP's suggestion of Petra, Jordan then Aqaba, Jordan and followed by Dahab, Egypt. You could try scuba diving in Dahab. Lodging is very inexpensive and you feel like you are in a different world. Its so unlike the Sharm el Sheikh, Hurghada Egypt. Dahab is very laid back and the restaurants are just cushions on the beach overlooking the sea. My husband who sounds like you heads to Dahab for diving anytime he is stressed.

You can get to SSH or CAI with *A. The flt CAI-SSH is less than an hour and the drive SSH-Dahab is about an hour. There is a ferry between Aqaba, Jordan and a town about an hour north of Dahab.

You can catch a glimpse of the culture of Dahab on youtube (Blue Hole). There are lots of beginner dive sites and other activities too.

If you decide to go, PM me if you need help with transport within Egypt because we know a front desk clerk who can help arrange everything. Have a great trip.