First, you must acknowledge that finding a group of sympatico players who are willing to play a game with a focus that is on drama (or even partially about drama) is going to take some work.
While it's true that not every game has to be about goofy fun, there are enough people out there who Embrace The Goofy (or what have you -- tactics, resource management, min-maxing) that you will probably not just stumble into a group that's gung ho to try it. A group that is largely composed of friends who have been gaming together since high school may indeed be All About The Goofy and have no interest in getting all serious on game night. For them it's about escape, and that's okay.
Finding out what they are willing to try is as easy as having a conversation with your group. This is an important part of Social Contract, a process that has gotten talked about a lot recently, and one that should be an ongoing conversation amongst a circle of players. If playing games is meant to be fun, you need to find out what your players enjoy, right? How do you do that? You ask them, straight up, what they want and don't want to see in the game. It's as simple as that. There's no pretentious B.S. involved -- it's a conversation aimed at making games as much fun as possible for all the players involved.
Maybe your players will be excited to try some drama as a side dish to their goofiness, or even as a new and spicy main course. Ask them.
The second step is a little tougher, especially for us nerds with our sometimes-fragile sense of social belonging. If your regular group is not open to the idea of adding drama to the gaming stew pot, or not many of them are, you're in the position of having to form a new group around a nucleus of players who are interested in giving drama a go.
Some people balk at walking away from their regular group to try something new, but this really doesn't mean you have to stop gaming with those people. You just don't play drama-centric games with them, full stop. You'll still be friends like before, and hang out, and do the same goofy things you've enjoyed for years. This is not an "either or" proposition.
Modern social media makes it a little easier to look for a new circle of players when you want to try new things. There are many groups on Facebook that cater to gaming in various cities around the world, as well as sign-up boards at your Friendly Local Game Shop, university clubs, LARP groups, even websites devoted to finding players. It may take some time, but it is definitely do-able.
When you meet your prospective new players, have open and frank discussions about the kinds of games you would like to play. Listen to what they have to say about the games they've liked in the past. Be willing to play games you've never thought of before, and look for people who are willing to do the same.
Once you've got some players who are willing to give a drama-based game a go, remember that it will take some time to build the trust and chemistry necessary for deep dramatic roleplaying. Don't worry about hitting it out of the park the first time, just work at getting those dramatic moments to happen and commit to them as much as possible. (Tomorrow's post will discuss my wife's most important contribution to the HTHD school of roleplaying, the "Fuck It" rule. YOU NEED TO READ THIS. Tune in Tomorrow.)
Time and patience pay off, when you're creating drama. If you're willing to put the work in, there is an opportunity to play games that are filled with the kind of satisfying, meaty scenes that the best TV dramas, movies, and books are capable of producing. Some people don't want that, and that's fine. But if you are someone who does...
Don't be a goof.