Congrats!
Options for deferment/forbearance on federal loans are spelled out at studentloans.gov. Private lender options are harder to figure out.
What you should be trying to get, imho, is a post-graduation 6 month grace period. You'll have to do the work to figure out if you're eligible, and to get the grace period to happen.
Thinking ahead, you cannot assume that you will be completely free of the requirement to make payments, as you transition to med school. You should assume that you will have to make some payments in the gaps between grace periods and deferments. Your med school money, and your med school deferment, may not come immediately - you may have to do work to get that deferment. And you are responsible (not your school, not the government) for pushing for that med school deferment.
So as you're planning your finances for the next year, make sure you're clear on what your med school money will cover, and what it won't. Med school money doesn't cover moving expenses, first/last on an apartment, car loans, credit card bills, or payments on student loans that come out of grace before your med school deferment.
I'm finding that many of my classmates in med school have parents etc who take care of these things. Many of my classmates haven't had to borrow a penny. My med school's financial aid office acts like it's a character flaw, on my part, that I don't have wealthy relatives. Point being: I have to be my own advocate in managing my undergrad debt, in addition to managing med school debt, and learning about post-med-school repayment.
Fact: you are responsible for your debt. Period. If you "simply can't afford" to pay your student loans, but you have a smartphone and a car, for instance, you are lying to yourself. You can lie to yourself about anything you like, but student loans would be a bad choice of things to be delusional about. Your lender, and the US Department of Education, are in charge of what you can afford. You aren't. The fact that you're starting med school next fall doesn't lessen your responsibility.
So, now it's time to educate yourself on your debt, so that you can make smart decisions that don't harm you during the next 10 years when you really, really need those deferments and forbearances. You have to own finding out what choices you have for the next year, and the consequences of those choices.
It's time to do inventory. What did you borrow, from whom, when, on what terms? Start with nslds.ed.gov, which will show you your federal debt. If you took any private loans, these are 100% separate, not tracked on NSLDS, not the same terms, and you need to be extra careful in keeping track of them.
"Sallie Mae" is a loan servicer as well as a lender, which means that "Sallie Mae" is not a good description of a loan. There are several types of private loans that SM has offered over the years, in addition to several types of federal loans that SM services for the Dept of Ed. So you have to do inventory within Sallie Mae to see what's in there.
Every time you sign a document, you need to keep a copy. If you did not keep a copy of a document you signed (whether you remember signing it or not) to get a student loan, you are vulnerable. Collect all your MPNs (master promissory notes) and all your notices of interest rate changes and such. Keep them organized in a safe place.
Best of luck to you.