As a former employee of Hilton, I can tell you that Hilton offers discounted rates for employees and their family members/friends alike. The friends/family rate is usually much higher than the employee rate. The documentation that you had sounds like the correct documentation for a family rate... Family rates get a print out on regular paper and employees get a 'passport card' that comes from Hilton and is made of cardstock. So while it sounds like everything was right on your end, it's just a misunderstanding at your destination hotels. I probably would have asked to speak to a manager.
IMHO, the manager in NM really has no control over the situation. It's the hotel's fault, not the manager. Hope this helps!