Can lymph nodes that are soft, moveable and do not grow still be cancer?

Less likely. Cancerous lymph nodes typically are hard. Lymphomas (another kind of malignancy in lymph nodes) might be softer, or even feel fluid-filled (cystic). Any unusual lymph node, one that increases in size, is tender, hard, clumped, fixed to surrounding tissue, etc needs to be checked promptly, especially if associated with other symptoms, like itching, fever, weight loss, jaundice. Any doubts, see md.
Stop worrying. If they're not growing, not fixed to nearby structures, and you don't have obvious signs / symptoms of illness, forget about cancer. All young adults have palpable neck nodes. Glad you're a good observer of your body.