-Why does the coasting rotating system slow down as water drips into the beaker?

You don't explain the experiment very well, but I'll suppose a beaker sits on a freely-spinning turntable as water drips into it.

As the total moment of inertia I increases due to the increasing water mass, the angular momentum h doesn't increase proportionally since the water is falling vertically and thus adds no angular momentum.

Spin rate omega = h/I, and with I increasing while h remains constant, omega decreases.

Coasting Rotating System

Weight of the water in the beaker increases so force of pressure increases friction therefore slowing the speed.

as water is rotating so while dripping water in beaker its effect on omega is as we know that omega=h/l so while dripping water h=constant and inertia increases with drips of water in beaker so omega decreases as it is inversely proportional to inertia. and hence rotational effect of water in beaker decreases.