I am curious as to how you are expecting $350k for four years of medical school - what does that number include in terms of expenses beyond tuition? Knowing that would help give you a better answer I think, but in general, I would suggest looking at the estimated salaries for your top 3 specialties. Similar to what has been said already, if you are looking to go into a specialty that won't make enough to justify the amount of debt you will be in, you will end up having to dig yourself out of that debt for decades. As much as you may love your job, having massive debt hanging over your head will be a huge mood killer. With that being said...
- If your preferred specialty makes good money, continue on your path and live as cheaply as possible going forward. Look at the median income for the area you settle on and build your non-financial-aid budget around that - but the less you spend the better as all of the excess can be sent straight to knocking down your debt.
- If your preferred specialty doesn't make enough money to justify medical school, look into employment related to whatever your master's degree was in, look for a less expensive school (in-state tuition?), or a mixture of the two. If you're going to school in NYC for the same reasons you want to move there post-medical school, you should consider looking at schools elsewhere. This will lower your costs during med school, and you likely won't have all that much time to see the family you want to be so close to anyway.
In either case, if you're set on staying in New York, why not live with the family that you are looking to stay close to? You mentioned it's not cost-effective to move, but given NYC is the most expensive city in the US to live in, it would be cost-effective to move just about anywhere else. However, if you need to stay near family, you should just save the big chunk of change and actually live
with the family if at all possible. If you can't live with your family and still want to live in/near NYC, find an area with lower rent/cost for living, even if that means a longer commute.
Really, you have tons of options to consider for your path forward, you'll just need to determine what you want to be doing, where you want to be, and how to make the numbers work out so you aren't drowning in debt for the rest of your life.