Hello Ben G, welcome to thegradcafe!
A year before applying to graduate schools, I contacted professors through email, introducing myself and expressing my interest to work with them. I clarified some things about the admission process and raised my concerns; I wanted to apply to MS/PhD programs in Neuroscience but I was (and still am) lacking hands-on experience on that subject as I have been working as a plant molecular biologist. Out of the five or so professors that I sent a message, only one responded. He was very kind and honest.
I did not end up applying to that university nor to a graduate program in Neuroscience but the advice that he gave was helpful. With regard to this whole "contacting-professors-before-application' issue, I suggest that you go ahead and send your target advisors an email. Write a message that will make you appear as a professional, knowledgeable on your field yet inquisitive and open-minded at the same time. If you don't get replies at first, don't get discouraged; as you have said, these are very busy people you are dealing with.
In addition, here are some links with the same topic: