As someone who works at a large, public accounting firm and is heavily involved in recruiting, perhaps I can give some advice.
Right now is prime recruiting time for public accounting firms for the coming spring and summer, so you're timing couldn't have been better. I work in the Southeast, so it may be different for other regions.
My only suggestion to your first point of sending resumes to partners is that you should ask for a business card and then follow up. Chances are if they are a partner/director, they will forget about you the minute they leave the building. That's not an insult to you, that is just the way they operate. If you follow-up, there is a much better chance that they actually take a look at your resume and forward it on to the recruiter or suggest that they interview you. Once you have gotten “in” with a partner/manager, request the contact information of the head recruiter. They will likely be the one to set you up with an interview/dinner, etc. However don't pester them. One or 2 calls to a partner at most and probably the same for a recruiter. If they don't call back or offer an interview, take the hint. If the seem nice, perhaps asked why you weren't selected for an interview. You may have been a great candidate, but they could have already filed the positions they needed.
You're idea of calling a recruiter is not a bad idea at all. Yes, they get a lot of calls, but that is their job. Have something prepared to say that shows you know a little about the firm. Nothing too corny, just a tid bit of info to show you've done your homework. Chances are you will be able to talk yourself into at least sending your resume and perhaps a follow-up interview. However, the drawback is that your resume is likely to piled in with other hundreds of resumes they already have. Similar to what I mentioned previously, a follow-up call will certainly help your chances. If you're going down that path, it would also help to ask if the firm has any upcoming events you can attend. Like you said, face to face is better than phone/email. (My personal advice, I wouldn't call 20 firms in one day. If you treat each firm as 1 of 20, your tone and conversation may reflect that. You want to treat each phone call as a separate interview and give your best effort. 5 a day maximum. 20 and you'll be tired/bored by the 8th call.)
I also suggest trying to attend a few career fairs. Ones at prominent Universities are the best (perhaps the one you attended!).
Also, use your resume as an advantage. Anything can be spun in a positive way. Even though most people jump from public accounting to industry, state that your experience will give you a unique and helpful perspective not many public staff accountants have.
Lastly, anytime you speak with a recruiter/partner/manager, show genuine interest. They can tell when it's fake. Be engaging, but not over the top. If you are friendly and can find a way to relate to the person you're speaking with, they have a much better chance of remembering you. Make friends, and then worry about getting a job. Sometimes the best thing you can talk about in an informal setting is anything BUT accounting.
Sorry if I came off like I was trying to give you social advice. I am only trying to give my perspective on recruiting! Good luck!