Considering the fact that Tufts is no longer need blind in their admissions process is exactly why you should be looking into applying to Questbridge forTufts. It does not matter what race you are, if you meet the income requirements, you will be eligible for Questbridge.
While no one can force you to do something that you don’t want to do, things will eventually come down to a question of worth. So far you are demonstrating that it is not worth it to take advantage of an opportunity that can provide you with a virtually free education.
So what you found out about the program late; this is where you grind it out put all of your free time in to getting the application done. This is also where you apologize for the short notice and explain to teachers that you just learned about the opportunity, explain your situation and the fact that it is one of the few options available for you to get an affordable education.
I would hope that you have garnered enough good will with 2 people who could write a recommendation for you (what they have to say about you, they don’t need 6 weeks to put together).