Husband feeling uneasy with rapid heartbeat, fatigue, hands swelling, shortness of breath. Fighting him to go to the er. Should he go to the er?

YES. These are serious signs of heart attack or arrhythmia - get him to ED asap!
Possibly. If these symptoms have developed very rapidly (ie in the last several days) I would say atleast an urgent care unless you can get an appointment with your primary care doctor asap. It's hard to know without more details if there is something very serious going on with him...But I would strongly consider seeing someone before the weekend.