When cat fur gets coated with cat skin oil, the fur can feel stiff or gluey. All that’s happening is grease + fur = stiffness. To fix the situation, you can use ball-tip scissors and carefully, CAREFULLLLLLY, trim off the oiled-up fur. The skin underneath will still be oily, so you can use a wet wipe made for cats/dogs. That will help a little.
If you’re feeling adventurous, you can use a diluted degreaser like Dawn or Palmolive — unscented original versions, diluted heavily — to wash the greasy skin and fur. Be SURE to rinse, rinse, rinse and rinse some more because you don’t want your cat to lick that soap.
A common place to find stiff, oiled-up fur is between the shoulder blades. I can’t believe how gluey and sticky this area can be. Very hard to degrease without washing a few times.
If fur is dry, that’s different. Could be that the fur was washed with a degreasing shampoo too many times in too short a time. More likely it’s not related to grooming but to some other situation. I’m not a vet, so I don’t know what would cause dry fur. I’ve never seen dried-out fur on a cat. I’ve seen oily fur many times.