How can I treat anxiety-related swallowing problems?

Meds and therapy. Anixety is a common problem treated by therapists and mds alike. Commonly prescribed anxiolytics such as prozac, (fluoxetine) ativan, and talk therapy that allows self-reflection can allow a person to come to terms with ones anxiety disorder.
See a psychotherapis. If you are sure the problem it is anxiety-related, then tackle the anxiety directly. See a psychotherapist with experience treating anxiety. It is definitely treatable, and i strongly suspect it is affecting your life in more ways that just swallowing.