Learn Spanish (Latin America): El hotel está completo.

It does, but this is one of those adjectives that changes its meaning depending on the verb! When used with estar, completo means "full" in the sense of "fully booked" or "at maximum capacity" (like a hotel, flight, or parking lot). If you used ser instead (el hotel es completo), it would mean the hotel is "comprehensive" or "fully equipped"—meaning it has every amenity you could want, like a pool, gym, and spa.

Could I say el hotel está lleno instead?

Yes, you can! Lleno means "full" in a physical sense, so el hotel está lleno works perfectly to say the building is packed with people. However, está completo is the standard, natural way in Spanish to say a business is "fully booked" or has "no vacancies."

If estar completo means "to be full," can I say estoy completo when I've eaten too much?

No, estar completo is generally used for places or vehicles reaching their booking limit or maximum capacity. If you are full from eating a big meal, you should use lleno (estoy lleno) or say estoy satisfecho (I am satisfied).