The main point is that the I-20 form is from the Department of Homeland Security and the F1 sticker is, like all visas, issued by the Department of State.
The F1 visa sticker in your passport is needed for the actual physical act of entering the country, and its validity has nothing to do with whether you're legally allowed to be in the country - that has to do with the dates on your (latest) I-20. This is why the stamp that you get in your passport when you enter the US has 'D/S' written on it, which means 'Duration of Status', meaning that you can stay in the US as long as you maintain your status as a student even if the sticker expires (but if you leave, you would still need a new sticker to re-enter the US). Conversely, if you stop being a student, you can't then use your F1 visa to enter the US even if the sticker still appears to be valid.
There is no way to apply for the F1 sticker without leaving the US (the department of state is fundamentally meant to work outside the US). It is highly suggested that you go through the process in your home country; technically it's not impossible to try in a third country, but there is a (small) possibility that they will reject your application as a citizen of a third country, unless you have a compelling reason for being there. Some US consulates, I believe, specifically say on their websites that they do not process visa applications from citizens of other countries.
What does this mean for you?
Don't worry, you'll have plenty of contact with the universities before they mail your new I-20. They are used to people in your situation, and they will ask you where to send your I-20. Alternatively, when you accepted an offer to a graduate program, directly contact the international students office and tell them where to send it, to avoid any controversy.
You don't need a new sticker to continue to be a legal resident of the US, as long as you get a fresh I-20. So you could have the university send you the I-20 directly, or have your parents mail it to you while you stay in the US. You only need to apply for a fresh sticker in your passport if you leave the US and wish to re-enter on the F1 status. (There is an exception for short travel to Canada and Mexico. )
Bear in mind that the sticker application process takes a bit of time. The break between undergrad and grad programs is probably the easiest time to apply for a new sticker. You can apply for a new F1 sticker even if your old one still has some time left in it. Probably good to apply sooner rather than later, given the state of US politics too.
The process for applying for a new sticker is the same as applying for a first one, except it might be easier since you've already had this visa and not caused any trouble.
There is a grace period where you are allowed to stay in the US between your undergrad I-20 expiring and your graduate I-20 starting.
You should confirm all of this with your local international students office, as well as the folks where you decide to go.
Best of luck with your applications!
PS: I think this is the right place for this question.
