Mucinex DM’s main active ingredients are guaifenesin and dextromethorphan. Dizziness is a known side effect of dextromethorphan with anxiousness occurring much much rarer. It is likely you won’t experience the anxiousness with one dose of Mucinex DM.
Xanax is a benzodiazepine used to treat anxiety. There are no known drug interactions between Mucinex DM and Xanax and it is theoretically safe to use the two together. But caution must always be advised when mixing drugs. Each patient is different and therefore check with your prescribing doctor before taking the two together.