If I get an Asian flush when I drink, does this mean I'm more likely to become an alcoholic?

Not likely. An "asian flush" is caused by a deficiency in an enzyme called aldehyde dehydrogenase. This interrupts a specific step in ethanol breakdown that allows acetaldehyde to build up, causing the symptoms you experience. This can be caused by certain drugs such as antebuse and certain antibiotics. Many people find this extremely unpleasant and curtail their drinking as a result.