Perhaps you do not realize what a gem Seattle is. Experiencing Seattle as a UW student is a lot different that as a townie. NYC is insanely expensive for almost everything. You have to send out for your laundry. It’s $30 to eat breakfast at a coffee shop. NYC, however, is wonderful with a generous budget.
To the $10K you’ll need to add $3000 in airfare, and 2-3x for other stuff like entertainment and food. No sports either.