Learn Spanish (Latin America): No deberías manejar borracho.

Using the conditional tense deberías (you shouldn't) makes this a piece of advice. If you used the present tense no debes manejar, it would sound much stronger, like a strict rule or command (you must not drive).

I learned conducir for 'to drive'. Is manejar different?

In Latin America, manejar is the most common, everyday word for driving a vehicle. You will still be understood if you use conducir, but it is more typical of Spain.

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

You would change the adjective to match her gender: No deberías manejar borracha. Even though it comes right after a verb, borracho (drunk) describes the person doing the driving, so it must agree with who you are talking to.

Why is manejar left in the infinitive (unconjugated) form?

Just like in English (shouldn't drive), when you have two verbs working together like this, only the first one is conjugated. Deberías already tells us who is doing the action and the tense, so manejar stays in its base form.