I'd like to tell you from my own experience that the position you are in right now will actually make moving out on your own much easier. Let me tell you why and how...
Stay with the grandparents for your first semester if you can, because you will need to concentrate on starting school (which is a big adjustment) and not on moving and getting set up. The less money you make the more money you get from financial aid. Everyone gets the same amount of money they have to pay back, but when you make less money from employment, you get more money in the form of grants that you will not have to pay back after you graduate.
Now, save up the money each of you have left over after paying your tuition and books...you should have several thousand dollars left over each semester. For example, me and my now husband would have a good $5000 plus left over EACH after tuition and that we would use to pay up all of our bills for the next 6 months until we would recieve our next aid the next semester. Only $1500 was a loan, a scholarship pain for my tuition and then the rest are grants. When the semester is over and you have time off for school break you take that saved up money and move out on your own and get settled in before it's time to start the next semester. Have the bills paid up and/or a little set aside as an emergency savings account (you will need it eventually, trust me!) and pay your expenses each semester and then you can concentrate on school and not having to work full-time. My husband would work part-time, maybe twenty hours a week, to help pay for gas or little expenses, because his classes were not as difficult as mine.
Keep your bills very simple...just the basics, no iphone with a $100 plus monthly bill. We pay utilities, rent, car insurance, used financial aid one semester to trade in our $500 dollar junker car we had to buy to get started at school for a more reliable car so no big car payments, we have one pay by the minute phone and the only "extra's" we have is a basic cable plan, internet we have to have for school and a cheap gym membership. We don't eat out, most of our clothes have holes in them and we live very cheaply.
Hopefully your grandparents can watch your child, because that can add a considerable expense, depending on where you live. In my area you can only qualify for child care assistance if both of you are working a specific number of hours, so we did not qualify for that. Instead we set up our classes so I would go to class the come home to watch the baby and he would go to work or night classes. It does suck to rarely see each other, and to live basically worrying about every dollar, but it will be worth it because we will graduate this semester!!!
Also, because you are not working, or working very little hours and most likely making minimum wage, you will also qualify for food assistance and health coverage for your little one. Take advantage of this, because soon you will graduate and become contributing, tax paying citizens and you will be paying back into the system off of the higher wage you will earn more than what you will be borrowing from the state.
And just an FYI, you do not have to be married to claim someone on your taxes as a dependent, so long as you are supporting them financially. My boyfriend claimed me and our son on his taxes because he was the only one working. You may be able to allow your grandparents to fill the two of you and take that extra money for themselves for helping the two of you out. If you haven't been working you and your girlfriend won't be able to file taxes for yourselves, because you haven't pain anything in to get back, you know what I mean. So let your grandparents claim you.