One of the things that haven't been mentioned (although I may have missed it due to being dopey) is your insurance. It may not cover you depending on the level of travel warning and your cover.

I would urge you to check this before you head out there, if you do decide to go. You don't want to be in a car accident and find your insurance invalid and having to pay a scary amount of money for repatriation even if you enjoy the place and find it perfectly safe.

As stated by Layz above, I also recommend Amman. I was happy to walk about on my own, I found people as well as cats wonderfully friendly, warm and kind. Not very touristy at all, with ruins and treasures that have not been rendered a tourist central.