I have the same dilemma too. I want to apply to the University of Tennessee... but it might be too late.
but, here's why we're different and why I think you should apply.
1. Chances are AADSAS already processed your application so if you add Tufts, that's just an extra mailing for them. I don't think it will take that much time for them to send out the stuff to Tufts. (In Tennessee, they have their own application and I'm not so sure how long it will take them to process it. Plus, I'm an out of stater.)
2. I don't think that Tufts sent out interviews yet.
3. From personal experience, my friend applied to Tufts over winter break last year (That's like december something) and she actually got an interview in March and eventually got waitlisted. If you apply now, you'd be way earlier than she was and probably have a greater chance than she did.
Hope I helped!