We were in a very similar situation, but living on the other side of the US. I was born in Canada. My daughters were born in the US. By US law they are US citizens because they were born here. By Canadian law they are Canadian citizens because I was born in Canada. It sounds like you are in exactly the same situation as my daughters. I believe that you are already a Canadian citizen – you just need to get the documentation to confirm this.
You need to get a “certificate of Canadian citizenship”. This takes multiple months. You should get the application sent in ASAP. In our cases the local Canadian consulate was very helpful in this process. However, you should get this started now if you are going to be attending university this coming September. What year of high school are you in?
Once you have a certificate of Canadian citizenship, you will want to get a Canadian passport. You need this by the time that you actually show up at the border to go to university. It is quite quick to get once you have the certificate of Canadian citizenship.
Then you look at Canadian universities and apply to Canadian universities. We found the application process very quick and straightforward.
One thing to watch for: A couple of universities in Canada missed the fact that my daughters have dual citizenship. Along with the acceptance they sent information about the cost for international students and how to apply for a student visa. A quick call to admissions plus a fax (of either the certificate or passport) fixed this quickly and easily.
And yes, you will pay the same tuition as a Canadian citizen (which you are). If you have never lived in Canada and if you attend McGill then you will probably get in-province tuition, which is even less. For us McGill is pretty close by. However, since you live in Washington state you probably would want to attend somewhere closer. UBC, Simon Fraser, and the University of Victoria are also very good and pretty close if you are anywhere near Seattle. There are many other good schools in Canada that we can suggest when the time comes. At least for us the cost of attending university in Canada is lower than the cost of our in-state public university, and this is even with a very good merit scholarship that we were offered by the in-state public university. The quality of education at universities in Canada is also very good.