Minor swelling around the wound site. Could it be a dry socket or is it just slow healing?

Normal healing. If this is in reference to a dental extraction, it is reasonable to expect minor swelling around an extraction site (wound). This is an acute phase inflammatory sign, and should resolve quickly in the absence of infection. If swelling persists or worsens, then infection or dry socket should be considered.
Probably normal. Minor swelling near a tooth extraction site is very common. A dry socket will present as intense/increasing pain typically 48 hours after a tooth extraction. It is caused by exposed bone in an area that has lost it's blood clot. Treatment is to have a medicated dressing placed into the affected area. Antibiotics will not help a dry socket.