You would always use "im" when saying that something takes place in a month ("im Januar", "im März" etc.) but that is kind of a fixed phrase. You would not say "in dem Januar/Februar/März etc.".
The months are not always pronounced with de definite article either. You could say for example:
"Januar ist ein kalter Monat." or: "Der Januar ist ein kalter Monat."
(In general you are right though, "im" is the combination of "in" and "dem".)
My answer was not accepted, "We go on vacation in February." I know its not a literal translation compared to "haben", as is "are on." However, "we go" is a much more colloquial way of saying this in America.