Did some of the passengers choose to go down with the ship?

Isador and Ida Straus

Yes. Near the end of the movie Titanic, we see an old couple embracing in bed as water pours into their cabin. The couple is first class passengers Isador and Ida Straus (left).

Isador was the co-owner of Macy's department store. In real life, Isador and Ida were both offered a place on Lifeboat No. 8, but Isador chose to stay on the Titanic so long as there were women who remained on the ship

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Ida refused to abandon her husband. Witnesses on the deck and in Lifeboat No. 8 heard Ida tell her husband, "We have been living together for many years. Where you go, I go." The couple was last seen sitting on a pair of deck chairs (not lying in bed like in the movie). Only Isador's body was recovered and identified.