Czipeter, I believe the 'inability' you have to 'do two things simultaneously' is actually a big part of any really good RPG, including AW. What's happening is this: Your interest as a player is to 'win'- to do well, to succeed at whatever it is the overall goal is perceived to be. Your character's interest, however, is to pursue their desires, dreams, and reasoning- sometimes at the cost of 'winning'. When you are asked to 'play your character', these two interests are in conflict.
Example of character interest trumping player interest: Dickinson and friends are fleeing a burning building. Dickinson sees the heavy steel door closing, and charges ahead to squeeze through right before it closes, shoving aside a wounded comrade to do so like the selfish git she is. The friends are trapped inside the burning building, and so are the documents they were paid to retrieve. Oops.
Example of player interest trumping character interest: Dickinson, a normally selfish character, holds open a heavy steel door long enough for the rest of the party to escape with the documents.. but not long enough for herself, so she dies in the conflagration. Oops.
The situation isn't always clear-cut, and sometimes you have to weigh one player's interests against another player's interests, or against everybody else's interests, and so on... but conflicting interests done well generates drama, which makes the situations interesting.
Some folks are used to this approach to role-playing, and some folks are not. Once you get used to it, though, it's a lot of fun. Hope it works out for you and yours!