I never had one of these cold-call situations work out- expect your mileage to vary. I chatted up nurses, sent letters, sent faxes + called, etc. The only one of these that worked out was a research opportunity, but I had actually met the lead research physician in an elevator already and he made the offer/suggestion first...so I was really just following up with a letter at a later time. Give it a try, for sure, but don't hesitate to fall back on the physicians you know because they'll probably give you better shadowing experiences anyway. They'll also be more likely to volunteer a letter of recommendation.
PS- I think the easiest path to getting "shadowing" on your application is surgery. Surgeons are completely used to having people observe them, and your presence isn't a burden on them in any way (they can pretty much ignore you). Shadowing in a primary care setting is different because the physician will probably feel more obligated to talking to you, and may have to explain your presence to every patient he sees that day.
