That is definitely not what is supposed to happen. Carmex is supposed to be hydrating, and the the protecting lip balm version with spf is supposed to decrease the likelihood of getting dry from the weather. If this is happening to you, discontinue use and try something else.
That is because it is medicated and meant for short term use. When you use it as a regular lip product it dries more than it helps causing you to add more of the product when that is just making your lips even drier in the long run