They actually met on the set of Password and he proposed on the show. When they got married they had to go back to work the next day (she was guesting once again) and at the end of it, they called each other Mr. & Mrs. Ludden - their way of announcing their wedding! I heard Betty tell the story twice and cried both times.
Another fact about Golden Girls - Originally Betty was cast as Blanche and Rue as Rose, but the chemistry wasn't right and they characters were too like their ones in Mary Tyler Moore and Maude, respectively. So they shot a second pilot (the only one shown) with the roles switched and it was magic! The way their characters played off each other was so unique and special.