There are evidence-based treatments such as progressive exposure therapy for your fear and every reason to believe that you can overcome this. Please look for a cognitive behavioral therapist with expertise in treating specific phobias. You can locate a therapist through the Association for Behavioral & Cognitive Therapy (www.abct.org) and the Anxiety Disorders Association of America (www.adaa.org).
Grieving is an important and healthy emotional step towards regaining her footing in her new life without your father, and yes, your mother's loss and bereavement could very well be developing into more of a depression. She could start with a check-up with her primary care doctor to address her emotional and physical symptoms. She might also want to consider participating in a bereavement group to be with others who have also experienced loss of their spouses.
Please do make sure that your mother has had a thorough medical examination to ensure that all of her symptoms have been addressed by her physician. In the course of that process, her medical doctor can be your ally in trying to figure out whether this is more of an emotional condition and can make referrals accordingly. You don't need to figure it out by yourself.
Management will depend upon the other symptoms and causes of her anorexia. Start with a thorough medical examination followed by behavioral health treatment, such as cognitive behavioral therapy to help her manage her symptoms. Medication will depend upon her response to behavioral intervention.