I have a MPP from Georgetown, and while I worked for 6 years before going back, there were quite a few people in my program that were straight out of undergrad. Having a great GPA and GRE scores helps, and several of them were econ majors (the program was pretty quant heavy at the time - now I think there are other pathways too).
I actually joke (but am half serious) that my time waiting tables and bartending is what prepared me the most for the international policy work I do now, because I have to deal with so many egos and difficult personalities. Being able to smile and come up with a solution that makes everyone happy is a skill I learned from those days, as well as being able to strike up a conversation with anyone. And there was one time that we were organizing an event to launch a report I wrote with several heads of state, and the green rooms were not properly set up, so I had to run through the event with trays of water and refreshments for the president of Tunisia. Definitely used my waitressing skills for that.
When I interview people for jobs at my organization, I always look for people who have some type of real "get your hands dirty" work experience. I've found that people who only have illustrious internships and fellowships make terrible employees, because no one has ever really depended on their work.
I guess what I'm saying is don't discount your experience, it can really be useful! Best of luck.