I have been experiencing what I believe are panic attacks. How do I get over them?

Face your fears . Do not avoid situations that make you anxious, rather, expose yourself to such situations on purpose. If you have a panic attack, don't do anything about it to try and get it to go away. That is called desensitization.
CBT. Medications can help, but counselling is also very effective. Cbt (cognitive behavioral therapy) can be very helpful at dealing with panic attacks. See your doctor and possibly a good psychologist.