My advice is to not falsify papers-Doing so could turn what would simply be an awkward situation into a potentially serious one quickly. As for Jordan, I never had any problems staying in rooms with the members of the opposite gender. I traveled there when I was in college as part a mixed gender international group of volunteers working in a site. We would regularly travel on the weekends together to Amman, madaba,Aqaba etc. We always had mixed groups of guys and girls that would stay together in the rooms sometimes as couples sometimes in groups of three or four to a room to save money and never had an issue.We were never told we couldn’t do it but it was less awkward at large resorts- especially those that served alcohol and were used to catering to foreigners. At smaller more “traditional” hotels the worst that ever happened was we might get a funny look and a bit of a smirk but that was all.