It basically costs me 60k for tuition and living. I have 60k saved after 10yrs. Should I knock out one year with my savings? Or use maybe 30k towards my savings? Or should I just hang onto it until I land a residency, then use it to start paying back?
In my opinion, what you are asking is if you should bank on one of the loan forgiveness programs.
If you are planning to do public service loan forgiveness or Pay as you Earn to pay down your debt, then the answer is no. Those programs will allow you to pay back a very small amount of your salary each year and then cancel your entire debt after 10 and 20 years respectively. In that case actually paying down your debt early is a bad idea. You should try and milk the stock market for a 5-7% rate of return and pay off as little as the programs allow.
On the other hand if you don't believe those programs will come through, you consider them too restricting, or you (like me) just think that six figure debt hanging over your head is a source of background stress that you would be fine paying to get rid of, then the right decision is definitely to start paying off this year's debt. Your debt is, for all intents and purposes, an investment that yields 6.8% a year no matter what. That's hard to match.
Whatever you do, I believe you should always maintain an emergency fund with 3-5K in it, and you should always carry 5K/month of disability insurance (costs about $100-$150 if its indexed to inflation). So it probably makes sense save 5 for emergencies, and save the other 5K for your $120/month disability payments, regardless of what you do with the other 50K
BTW I'm assuming you're going to a US non-profit medical school when you ask these questions. If you are going to a for-profit or foreign/Caribbean school the correct answer is not to go. They are trying to scam you out of your money. Also I'm assuming that you're single, not coming up on retirement, and know enough to carry good health insurance.