Learn Spanish (Latin America): Tengo los labios secos en invierno.

In Spanish, when talking about body parts, clothing, or personal items, you usually use a definite article (el, la, los, las) rather than a possessive pronoun (mi, tu, etc.). Because the verb Tengo already makes it perfectly clear whose lips they are, saying mis labios sounds slightly unnatural and repetitive to a native speaker.

Since this lesson is about augmentatives, how would I change this sentence to say I have big, dry lips?

You can add an augmentative suffix to labio. The most natural one here is -ote. Drop the final vowel of labio and add -otes to get labiotes. Your sentence becomes: Tengo los labiotes secos en invierno. Suffixes like -ote and -ón are frequently used with body parts to emphasize their size in a casual, subjective way.

In English we usually say "in the winter." Why is there no el before invierno?

Both en invierno and en el invierno are grammatically correct and widely used. However, it is very common in Spanish to drop the definite article after the preposition en when talking about seasons in a general, recurring sense like this.