Bones and Parenting not working properly in Blender

Hi, so, I have this problem, where I parent the bones, and I want to keep them offset.
It’s a hand. In Edit mode I first select each of the 5 fingers, and then I lastly select the hand’s bone. I use CTRL+P, and select “Keep offset.”

Now the problem starts. When I go into Animation mode, and I want to rotate any of the finger bones, they just keep moving about, and their head does not stay in place relative to the parent bone.

It works normally if the bones are connected. Is there a way to fix this? I don’t want to have more bones than necessary on my armature. Help!

First, have you resized the armature and, if so, did you apply those changes?
Second, uploading your .blend file will take a lot of the guesswork out of diagnosing the problem.