Why in the world would you purchase Oak to paint? Did you tell the retailer that you wanted to paint them, if you did and they sold you oak cabinets NEVER repeat NEVER go back there!
As other post stated you should have purchased birch, Or Alder/Poplar frame and MDF doors.
What did the painter paint them with, oil or waterbased paint? My guess since they are fuzzy is that he used waterbased and/or didn't sand before painting.
Yes you can save them, It is going to be a lot of work. You need to sand the doors and skim coat with filler, spackle, Autobody spot putty, will work. Do NOT use wood filler. Sand, Skim coat, sand and reprime.....if you can see it in the primer YOU WILL see it in the top coat.
Use a high quality acrylic primer, Let dry overnight and sand lightly, if you need to fill a spot do it and reprime.
Sorry for the shouting, I make cabinets and remodel for a living. It just pisses me off when someone makes something so easy so hard.