Both nipples have been erect (without going down) constantly for 7 days straight & nipples are sore (though my breasts aren't. What could cause this?

Not entirely sure. I will assume that you are a new mother who is breastfeeding. Nipple engorgement can occur with accumulation of excess milk, which can be associated with pain and tenderness. It is managed by removal of the milk. Another cause of nipple erection is sexual arousal or use of a nipple toy, although these shouldn't account for persistent erection for 7 days.