According to the standard blood compatibility chart:
(image from Owlcation: The Basics of Blood and Blood Typing - Wikipedia file)
AB blood types can only donate to other AB blood types, but AB blood types can receive from other blood types, so what is the point of being an AB donor.
Well, the chart says that they are the only blood type that can do a full blood transfusion to other AB types.
Also, organs aren't that easy to find - there are more people waiting for an organ than there are organs.
Also, the Wikipedia article on blood donation says that this chart is only applicable to red blood cells. For plasma and platelets, the situation is reversed.
Type O negative is often cited as the "universal donor" but this only refers to red cell and whole blood transfusions. For plasma and platelet transfusions the system is reversed: AB positive is the universal platelet donor type while both AB positive and AB negative are universal plasma donor types.
As for what happens between an organ donor dying and their organs being transplanted,
organDdonor.gov has this lowdown:
While the search for matching recipients is under way, the deceased donor's organs are maintained on artificial support. Machines keep blood containing oxygen flowing to the organs. The condition of each organ is carefully monitored by the hospital medical staff and the OPO procurement coordinator.
A transplant surgical team replaces the medical team that treated the patient before death. (The medical team trying to save the patient’s life and the transplant team are never the same team.)
The surgical team removes the organs and tissues from the donor's body in an operating room. First, organs are recovered, and then additional authorized tissues such as bone, cornea, and skin. All incisions are surgically closed. Organ donation does not interfere with open-casket funerals.
Organs remain healthy only for a short period of time after removal from the donor, so minutes count. The OPO representative arranges the transportation of the organs to the hospitals of the intended recipients. Transportation depends on the distance involved, and can include ambulances, helicopters, and commercial airplanes.