When you do feel that way it is really important to empty your breasts fully - even if you can't locate the blockage.
The usual process is hot pack/feed baby/pump/hand express/cold pack - because usually the blockage will be the very last bit to clear.
And taking ibuprofen reduces the swelling and makes clearing the blockage easier - in addition to helping with the pain.

The physio treatment works really well, but does require you to know where the blockage is - they actually work on the particular area of the duct involved. You don't need a specialist women's health physio, but you do need one that has experience in duct-zapping.

When you get really engorged, some of the stuff in your milk enters back into your blood, and provokes an immune response so you feel yuk - even before the infection that is characteristic of full-on mastitis. And at that point unblocking the duct can still fix the problem. But really simply, if the antibiotics fix it then it is mastitis.