Wrong you can have mutiple colds in the same winter. You can expell one cold and catch another. We can get it the winter after. There are so many strains of influenza - The common cold that you can catch a cold whenever. Colds are not always caught in the witner. The viral particles would not die because the infection can pass through blood. I.E blood splatter.
I concur with what Brambleclaw said. Firstly, the common cold can be caught at any time throughout the year. In fact, it's been noted that the responsible viral infective agents - the
Rhinoviruses - are quite common in Spring (especially), Summer and Autumn, in addition to Winter. Secondly, it's virtually impossible to cure a common cold. Due to hundreds of mutations and adaptations, there are many different strains and types of the common cold. In fact, 'common cold' is merely an umbrella term for a plethora of different diseases, yet all have the same symptoms, more or less. (The killer disease tuberculosis also seems to follow a similar pattern.)
In application to the topic, if a
Rhinovirus structured exactly the same as one which carries a strain of common cold were to be the carrier of the zombifying disease and mutation were to occur, it would be very hard to find a vaccine, let alone a cure (not impossible, though, but it would prove quite difficult.) In addition to that difficulty, however, even if a cure were to be developed - what about administering it? It wouldn't prove to be easy to catch and cure a zombie (Yes, yes, Will Smith in I Am Legend seemed to do it, but that's another story). But what about the tens of thousands who could become infected? And then there's still the amount required...
But, hey, on the positive side - Who needs a chemical cure when a shotgun round to the head will do?

And, unlike a chemical cure, you don't need to catch, restrain and, presumably, inject. Simply, get a nice, close range, cock the shotgun, and press the trigger. Instant problem solving. XD