Generally speaking, there’s no need to accept an offer as soon as you receive it. Offers have a standard deadline for you to respond. It is normally somewhere on your offer letter. It’s absolutely fine to make further inquiries with HR or one of your prospective advisors regarding funding opportunities:
Can I apply for financial assistance?
Are there any scholarships or teaching assistantships I qualify for?
Can I be funded by your research grant professor X?
All excellent and valid questions that show your genuine interest in enrolling. Graduate school is expensive, and not everyone has the financial privilege of spending several years with no financial support. Admission committees know this and will try to make it easier for good applicants to get some support if possible.
Accepting an offer and then backing out in favor of another is a big no no. Not only did you waste everyone’s time, you also made some other candidate miss an opportunity. People will rightly be upset.
What you can do is ping the schools you’re waiting on. A simple “Hello, I just received a few offers I’m seriously considering and would like to know whether your department had the chance to review my application. I need to make a final decision by XXXXX so would appreciate an answer before that deadline, many thanks!”
You will likely get an answer pretty fast.
Edit: as Alexander Woo mentions in their answer, taking up an unfunded PhD is a bad idea. You may find yourself in a financial hole that you may not be able to get out of for years to come, having to hold off on other important life decisions (buying a house, starting a family) that may look very far away now, but will become frighteningly, depressingly real before you know it. This comment does not hold for Masters degrees: these are often offered on a full tuition deal, but are also a short term commitment.
