It will be difficult to get around the issue of getting a US license for insurance on a vehicle by switching insurance companies every month or so. It is not easy to get insurance with a foreign or international license at first and then if you begin to cancel insurance every few weeks it will show up when you are trying to get new insurance. Insurance companies will likely see this as a red flag and not want to insure you.
The DMV usually keeps tabs of insurance polices, when they cancel or lapse, to make sure a car is not without insurance. Changing insurance companies so often might cause the DMV to become suspicious. Look into getting a Washington state license if you want to buy a car, it might not be as difficult as you think. There is information at www.dol.wa.gov/ds/dl.htm that should help you get started.
Once you have the car and license you can get a WA free car insurance quote.