Thanks for all the comments everyone. I've given it a lot of consideration and I've decided to defer for a year and try to pad my resume further. If I don't get a fellowship at that point, I've decided that I'll likely decline.
On the subject of padding your resume: the key issues for determining whether you get funding or not are likely to be: 1) GRE scores, 2) recommendations, and 3) your ability to articulate a clear and compelling research agenda. In a department with a shortage of funding, this last is likely to be the most important in determining whether you get funding. It may be very important for you to become familiar with the various funded research projects of the faculty, so you can see whether you might be a "fit" for one of them. In many cases it can be beneficial to establish communication with faculty ahead of time to determine whether they will be accepting new grad students for the coming year, and whether they are looking for research assistants.
Good luck!