Well, the way I see it- you’re not that far off.
2014-2015 Tuition: $48,700 (sheesh, that’s $9K more than I paid)
Room and board: $11,700 (this varies depending on the dorm/set-up)
Books/supplies $1,275
Fees: $410
That’s all you have to pay for school itself (we’ll talk about transportation and stuff in a bit). Total: $62,085. That’s $16,000 more than your package. If you count your family’s $10,000 contribution you’re at $6,000. If you’re able/willing to work, it’s very likely that you can make that much money over the course of the year (especially if you take into consideration a summer job). If you’re worried about working and finding time to study, have no fear- there’s a lot of jobs you can find ON CAMPUS that’ll be low-demanding enough so that you can study while you work. I had friends that did just that.
If you have a job right now or are willing to get one after you graduate HS, you should be able to put aside a few thousand $$$. This will cover your personal goods (bedding, towels, clothes, alarm clock, etc) if your family wasn’t planning on paying for this already. You should also have enough left over to pay for your own flights, at least for beginning of school year and Christmas (if money’s tight and your family is in CA then you might have to sacrifice a few of those extra trips around Thanksgiving and maybe spring break. Maybe, depends on when you can find and start a job on campus).
So, if this sounds like something you want to pursue, I think the way you’d go about it (you’d have to check with either Admissions or Student Accounts, I forget which) then you should be able to set-up a payment plan for the remainder of the money you owe the university. That’s how you’d be able to pay that $6,000 as you go ($6K over 9 academic months is only $666 a month. That should be very easy to earn at any on- or off-campus job).
Even better, if you DO decide to go for it and work through your first year, there’s a chance you won’t even have to work during the school year again (unless you want spending money). If you can get into the Resident Adviser program (http://studentengagement.gwu.edu/resident-advisors) you get FREE room and board. I have no idea how competitive the program is, but if you can swing it you’d be knocking-off $10-14K a year from your bill. That way your family’s contribution can go to books and transportation 
Ok, I said a lot, lol (the caffeine is really flowing this morning). BUT, if GW is something you REALLY want it can work. You’d be working through your 4 years (I did it, most of my friends did it) but assuming your FinAid package stays the same, you’d be graduating with around $30K in debt (less if you can get free/discounted room and board and apply your monthly income towards your school bill). That’s more than the $20k you want to spend but you’re still at the national average (I highly recommend staying at or below $30K in total debt).
Caveat: I’m not a financial adviser or anything like that, but I know the school and I’m the kind of person who figures out how to get what you want (I had to fight to go/stay at GW). If you have any questions about the above feel free to ask. You still have plenty of time to figure out if this is something you want to pursue, and I’m glad to help someone achieve their goal of attending GW