Quote:
Inside the tank is the type of float that rides up the water column, NOT the arm on top leading off to a float ball.
You have a FluidMaster ballcock,(see image) the seal needs to be replaced, (see image for pictorial) Now for the slow fill: Sometimes the orifice in the seal will swell shut and slow the volume of water entering your tank. This is a common problem with these ballcocks. They have a small washer with a hole,(see image) in the center that swells shut in time preventing water from entering the tank.