Contacting prospective advisors before graduate school applications - courtesy or annoyance?
Actually, it can be either a courtesy or an annoyance, depending on a number of things. A cold email from a student in an unrelated field that looks like a mass mailing is clearly annoying and will be discarded without an answer. But an initial contact from one of your professors with a recommendation to look at you, especially if they are colleagues, will meet with success. Most contacts are somewhere between these extremes. 

I think you were lucky to get responses from both. The fact that neither was dismissive suggests you had a good approach. 

The prof who refused you, I suggest, was being honest. Others were in the pipeline already and there was little time, as you say, to make a good and fair analysis. 

I don't recommend sending out resumes blindly to a bunch of folks. It is unlikely to result in any success. But when you are stuck and in a time bind yourself, you do what you must. But having a faculty advocate initiate a conversation in your case might have been an even better way to proceed. 

And, as always, do your homework. If your background and interest are aligned with those of the recipient you stand a better chance. Even better if you can indicate that you know something about their work. Blind praise of their work, not knowing what it really is, is easy to detect and such letter easily slide into the trash.