So, checking the COA of the various schools and including scholarships, it looks like OP would have to pay the following:
$26,000 per year at Rutgers
$35,000 per year at UDel
$45,000 per year at Tulane (plus whatever the costs are to get home)
I don’t think you’ve given us a clear answer on what exactly your parents can afford without going into quite a bit of debt. Since you didn’t receive financial aid anywhere, I assume they can afford some amount per year, but you need to sit down with them and discuss these issues. While I understand that you might be willing to have them go into debt (because that is what will happen here -they will have to cosign or sign for most of the loans that are taken out) and they might even say they are willing to do this, part of maturity is understanding that you might need to make sacrifices for them, just as they are obviously willing to make sacrifices for you.
I happen to think that U Del is an excellent school and the area is quite safe. It has some strong programs and if affordable, and you absolutely hate the idea of Rutgers, I would have no qualms about attending U Del.