I will be applying for post-docs as well. I applied for faculty positions, but I haven't heard anything back, so I am thinking that I am not quite competitive enough without a post-doc. Not all is lost, however, I do look forward to gaining a bit more experience and getting some more manuscripts out before the tenure clock starts ticking.
I think the number to apply to is difficult to know. I am applying to eight, which I think may be a bit on the high side, since I hear research post-docs are easier to get than clinical post-docs, and most of the post-docs on my list are research-oriented. Luckily a few of mine give offers before the uniform notification day, so hopefully I can figure out where I am going before then (since sites will often tell you your status if you have a competing offer).
Good luck, everyone!