Could a fibromyalgia pain come and go. If I rest I have very little pain but if I do my daily chores I feel some pain in my shoulders&lower back.?

Focus muscles. Fibromyalgia iis a chronic pain syndrome. It can be made worse by poor exercise, poor sleep, poor control of stress and psychologic issues. Each of these need attention. The pain is in the muscles and the muscles need exercising and conditioning. Being deconditioned is being out of shape. The better your muscles do, can make a big difference. The pain can vary with good days and bad.
Yes. The chronic widespread body pain of fibromyalgia syndrome [fms] can be relieved somewhat by rest and can become more symptomatic with physical activity. Despite that, in the absence of active tissue injury, physical activity is important to maintenance of heart/lung function and physical fitness. The low back discomfort could relate to a comorbid myofascial pain syndrome in the iliopsoas muscles.
Fibromyalgia. Yes, movement can cause the muscles to have spasm. You may want to consider some causes of the spasm such as disc problems of your neck or other illnesses such as fatigue illnesses : thyroid, anemia, depression, sleep problems, etc.