One nipple suddenly seems larger and I notice visible veins on that breast only. I'm only 39 and am not pregnant. Should I get a mammogram asap? Cancer?

You should. first see your doctor for an evaluation. A mammogram and/or breast ultrasound may be indicated.
Yes Get a Mamo. While this is unlikely to be breast cancer, new onset breast asymmetry should be evaluated by a physician and a mammogram would typically be part of that assessment in conjunction with a physical breast exam. .
See dr. You should see the dr and get a breast exam and a full clinical evaluation which may include the ordering of some diagnostic testing. Best wishes to you.