Learn Polish: Ubierz się ciepło, inaczej będziesz chory. - Dress warmly, otherwise you will be ill.

The verb ubrać się (to dress oneself) is reflexive in Polish. If you just said ubierz, you would need to specify an object, like a child or a specific piece of clothing. To simply say "get dressed," you must include się.

Does inaczej always mean "otherwise"?

It is used as "otherwise" to introduce a consequence, just like in this sentence. However, it also translates to "differently" when used as an adverb describing an action. For example, Zróbmy to inaczej means "Let's do it differently."

Why is chory in the nominative case? Doesn't the verb być (to be) require the instrumental case?

The verb być (seen here as będziesz) takes the instrumental case when followed by a noun, such as Będziesz lekarzem (You will be a doctor). But when it is followed only by an adjective like chory, you must use the nominative case.

How would this sentence change if I were speaking to a woman?

Only the last word changes. Because chory is an adjective, it must agree with the gender of the person you are addressing. For a woman, it becomes chora: Ubierz się ciepło, inaczej będziesz chora.