Are anxiety and panic attacks as common when you are drunk as they are when you're high?

No. Alcohol is a depressant, so slows things down--unless your trigger happens to involve alcohol, anxiety/panic should be somewhat less. Most drugs of abuse tend to speed you up (coc/crack/amphet/xtc/g/etc.), and they can increase underlying anxiety and make things worse. Neither is the answer--work with a quallified therapist to address triggers and improve overall.