I've got radio silence after applying to a ton of judges and some public interest places (not even trying for any paid or prestigious positions) about two weeks ago. They don't know my grades. I'm still applying to more places, but I'm not getting my hopes up. I'm clearly close to the bottom of my class, there's nothing too special about my resume, and I don't currently have any connections I can work.
Should I start begging professors to let me do research for them (would they even let me)? Would working in the law library be acceptable? Should I consider some study abroad thing just so I'm doing something?
What does your Career Services Office say? You go to a Top 10 school, so although you might not PERSONALLY have connections, you should have a great alumni network to tap into. Ask your professors for suggestions, and yes, see if they need summer research help. Working in the law library probably won't cut it.
If you're willing to work for free, there should be absolutely no reason why you can't get legal work this summer. Be proactive- contact public interest orgs or government agencies that don't have formal internship programs and offer your help. Include your references and writing sample, and if that's enough to get you an interview, they may never ask for your transcript.