In the kitchen with Phyllis, Angela begins putting her hairnet on. The ponytail she has is clearly still sticking out of the back. The shot changes and, one second later, the hairnet is perfect, and her ponytail tucked underneath.
It is unlikely that Meredith would be so opposed to admitting her "problem" as she was very open about it in Boys and Girls (2006), where she nearly admits to being an alcoholic and later declares that in 5 years she wants to be 5 years sober. Both of these indicate she has already enrolled in Alcoholics Anonymous.
ETA: It is not unlikely at all. She has never admitted openly that she is an alcoholic, using that word. Even if she is or has been in the AA doesn't mean she's ready to admit it to her co-workers. She is put on the spot by Michael and she is also inebriated, i.e. not in a great spot to admit vulnerability. That might explain why she denies being an alcoholic.
ETA: It is not unlikely at all. She has never admitted openly that she is an alcoholic, using that word. Even if she is or has been in the AA doesn't mean she's ready to admit it to her co-workers. She is put on the spot by Michael and she is also inebriated, i.e. not in a great spot to admit vulnerability. That might explain why she denies being an alcoholic.