Oberlin, no question.
I read that Drew may be experiencing financial challenges. Best of luck to you and your family. With their encouragement, Oberlin sounds like a great choice.
Go to Oberlin and promise yourself that you’ll text, email or Skype every day. It’ll be a difficult promise to keep at times but it’ll make your parents’ day and they’ll like that much more than if you were just at Drew.
Oberlin will offer you so much more. There are several flights every day between Newark and Cleveland, so you can get home quickly if necessary. While I usually suggest that people evaluate a lot of things from their own perspective, I think that you should pick Oberlin.
That cost difference will cover a lot of flights home, and/or a car. Oberlin is much stronger for your interests, and it’s not SO far that you can’t get home if you need to. There are so many more resources backing what Oberlin is able to offer you - the college’s endowment is five times the size of Drew’s, and it has a Forbes financial rating of A, to Drew’s C (as per comment up-thread about financial troubles). This is a great offer that outweighs convenience or a relationship with a single professor. I’d go with Oberlin. If you need to take a LOA at some point, to be at home with your dad, you can still make that decision if/when the time comes. Establish yourself at Oberlin, keep in close touch with your folks ad MYOS said, and go from there.
(P.S. Salary data needs to be viewed with a VERY critical eye. A lot of Oberlin students are probably in grad programs 4 years out, plus there are all the Conservatory grads who don’t have a steep ramp-up in salary as musicians. There’s no rational argument that Oberlin will disadvantage you earnings-wise, as an individual, compared to Drew.)
I personally would ignore the “average starting salary” data. This is largely based on what major people choose. For example, schools with large computer science and engineering programs will have higher starting salaries on average, but this will not help people with other majors.
“I … NEED to go to/be able to afford graduate school.”
This is very important, and shows that you are thinking this through.
I do not know what the financial state of either university is. However, given the offers that they have made, to me your obvious choices seem to be to go to Oberlin, or to call Drew first thing tomorrow morning and see whether they will match Oberlin’s offer (and go to Oberlin unless they can).
Academically I think that you have two good choices here. Having a good program and strong professors in your field does significantly offset the overall reputation of the school. Graduate admissions will know about strong programs at a very wide range of schools.
As a parent, I can only echo what others have said. – Oberlin, without a doubt, is where I’d be pushing if you were my kid.
In the age of Skype, I’d feel I’d see you.
As a parent, I want the best for my kid. Not just as a prestige factor, but the best for their major and meeting their dreams.
As an adult, I want the least financial pressure.
Per a request for an update, I’m excited to be attending Oberlin this fall! I am beyond appreciative for all of your help and I thank you all for your input and well wishes for my father’s health.
Congratulations on a great choice - I hope you have a wonderful time at Oberlin. Best wishes to you and your father.
Congratulations!!
Why would you even consider going to an expensive Private school in your situation? There’s nothing particularly prestigious about either Drew or Oberlin either. Did you apply to tcnj or Rutgers? How about University of Delaware which is fairly generous with merit aid? People on here will differ but in my view is a waste of money in this economy to major in history. You should change your major to a stem field or business.
Wow @njdadjets super not helpful. Did you read the thread? Nothing in your comment is responsive to OP’s initial question or situation.
Congratulations! I hope your father responds well to treatment.
@njdadjets is on a tear, telling every aspiring humanities major to study stem or business. “Today’s economy” is one of manufactured scarcity, and the people who are profiting from that would be thrilled if the study of history were abandoned. We need historians and the wisdom and perspective they generate, now more than ever. Oberlin is in excellent school, and you get to go there for under $10K/year - cheaper than your in-state publics. Go do great things and have a wonderful college experience. 
@njdadjets OP will be attending Oberlin for less than 10K per year on financial aid. I helped a Pell eligible student in NJ and all offers from 4 year in-state publics came in higher than that.
History majors learn how to read and write critically, to analyze, and to discern between reliable and questionable sources of data and information. Those skills will always have a place in any economy.
@njdadjets - most unhelpful post I have seen. FYI, this is from U of Delaware “Our aim is to make a UD education affordable to all qualified residents of our state.” Unless OP is from Delaware it is not very likely he/she would have received FA from U of Delaware. More often than not, private schools are more likely to give FA than public schools. Most public schools only give aids to residents.
Generally I would pick Oberlin for better prestige and cheaper net price…but how does it fit you?
Would you rather be the “it” kid or be challenged by other peers?
OP has made his/her decision. Congratulations!!!