Must the full name be used in the greeting of a letter to a college professor (from a student) or is it correct to write simply "Chère Mme/Cher M.?
First of all, do not use “Cher“ or “Chère”, this should be used for friends or relatives.
I would use “Madame,” or “Monsieur,” without any name or title as, academics rarely are referred to by titles (“M. Dupont” could have a PhD and a professorship, he would still by “M. Dupont“, not “*
You could use “Monsieur le professeur,”, but it sounds heavy and a bit outdated to me. And the feminine is kind of a mess currently: some expect “Madame le professeur,”, some others “Madame la professeur,”, some want a feminine form. In that case, *“Madame la professeure,” is sometimes use, but I would argue in favor of “Madame la professoresse,”.
In any case, neither “Madame” nor “Monsieur“ should be abbreviated unless it is in front of a name. “Mme Dupont” is correct, but “*
You should not use their name but their title :
In France (I don't know about other French speaking countries) using the name in letter greetings or when adressing s.o. is not usually considered as correct in formal language. Nevertheless it has become very popular over the last 30-40 years as a result of contamination from English, first used by the media it has spread among the general public, whether in written or oral form.
A few links to get help with letter writing :