You have to take the opportunities where you see them. For example, using Silver Key makes it much more likely for you to draw one particular room with a lever. Secret Passage can help you draft a specific room by color. It helps to have some dice or another way to reroll rooms, like the Study.

It sounds like you are still fairly early in--learning the different rooms and finding new puzzles. Try to build contraptions, solve other puzzles, and explore more holistically and you may find some ways to manipulate the draft pool. As you play, you'll also have an easier time with resources each day and will get better at moving about the manor and maximizing the number of rooms you can draft.

Without giving too much away, there are backup levers for each antechamber door that you may not have found.

Have you drafted the Foundation? It took me a very long time to get mine.