Let me introduce you to a tool called Drama Time.
Come up with a term that will mark a switch in your game into intense storytelling mode. I call it “Drama Time.”
Discuss this with your players. Explain your desire for deeper role-playing. During the game, when you feel it’s appropriate, announce “Drama Time” and hold your players to it. If someone starts joking, frown at him or her, and continue with the immersion. If players start to derail, say “Come on, guys.”
Does that require a firm approach on your part? Yes. This is a situation in which, if you want to change player behavior, you have to be the alpha dog. You have to be firm–not loud or harsh, but calmly definitive–about the kind of behavior that’s inappropriate.
Of course, don’t expect perfect adherence starting at the first moment. But when I’ve used Drama Time, players quickly grew serious and worked to immerse themselves.
Worth a try.