Drama here means "high emotion." It is a slightly separate idea from the category of "drama" television / plays / movies. Ex. - "You cause way too much drama!" (you cause too much emotion) Ex. - "Don't be a drama queen." ("drama queen" being slang, for someone who causes drama...someone who causes emotional or unexpected events to happen).

To be emotionally evolved means he has a highly developed sensitivity to the emotions involved in any given issue or incident. That helps him see more clearly the matters he has to deal with. Perhaps precisely because of this clarity, he has chosen to avoid a dramatic approach to solving problems, dealing with people and reacting to events. To avoid drama is to avoid arousing unnecessary emotions. It does not mean he is cold or distant. It is a matter of giving human emotions their due place but no more. It refers to a sense of proportions and not to the total elimination of emotions. "Ethos" means the characteristic spirit of a way of doing things or way of life.

"Drama" can also mean being dramatic in real life. Getting into fights, for example. If you are in high school, and you and your best friend get in a long argument, which lasts for several weeks, that can be referred to as drama. What the article means is that Obama is not engaging when people say things about him. He is attempting to remain rational, without responding in ways that may be poorly-thought out or emotional. For example, he hasn't directly responded to the fact that the Republican party plans to sue him, because he doesn't want to get involved in the drama around that.