No, you cannot be sued for not showing up to take on the responsibilities of your job. However, your family is right and if you are going to live in this world, you will soon learn that a Dairy Queen job is going to be the easiest job you will ever hate.
If you are too shy to do it in person, then pick up your cell phone and talk to the manager and let them know that you are quitting the job for personal reasons. Hang up and be proud to have done the right thing.
If you cannot handle a job in a Dairy Queen, then you may want to seek out some psychological help along with a trial period of medication. Because unless you can get out of your own way, you will be tripping over your problems for the rest of your life. Don't do that to yourself. You NEED to put the work into being a productive member of society IF you want to have a happy life and an income to pay for it.