From the Breastfeeding Answer Book, p. 493: When the breast is engorged, the blood and lymph move slowly through the breast tissue, allowing some fluid in the blood to seep into breast tissues, which causes swelling which causes tenderness, warmth, pain, and even fever in some moms.
For engorgement to cause fever, it must be able to make a woman's body *think* it needs to fight off some invader. Another post commented on the possiblity of your response being an auto-immune response - which sounds quite likely to me - your body may be over-reacting to its engorgement. I am not sure if there is anything you could do about this. Sorry, this doesn't help you!
Another idea is that whatever is making the tissues swell with fluid may be making the tissues of your throat swell (with fluid?). Again, not sure if there is anything you could do about it.
When it comes time to wean, you will need to do so slowly - removing one feeding at a time. This can be a natural progression, so you may not even notice any engorgement if your child gradually cuts back on nursing. Another idea is looking into substances (natural or prescription) that reduce your milk supply. But make sure it is safe to take while still breastfeeding!
You may never know exactly WHY this happens to you. But at least you won't be breastfeeding forever, so the problem *should* go away after you wean. If it doesn't, that means it is something more system-wide than just related to your breasts.