I'm a new grad, too, and I applied to a lot of places and got the standardized, canned rejection email from a lot of them. I have a full-time job because I was offered one at the hospital that I currently work at as a per diem CNA. So I'm lucky. I'm sure I could have found a job, eventually, at a nursing home or if I extended out my search radius, but I'm definitely glad for the opportunity I've been given.
That being said, I'd definitely take the part time job. A job is a job, and a part-time job is better than no job, at all. When I was a CNA, I had gone for a full-time position at my hospital and was given per diem because a former employee had come back and they gave the position I applied for and interviewed for to her instead. Thank God I was because it worked better for my schedule, but it also meant I could pick up hours whenever I wanted. Take the part-time. Hopefully, they'll give you a full-time orientation. And when you can, pick up extra shifts, see if you can meet full-time hours just by picking up. And then when there's an opening for a full-time position, go for that. Part-time, at least, gets you a foot in the door like other people have said. Sometimes getting jobs is all about the networking and already having an in.