I make my own drippers. Try replacing your valve with one meant for aquarium air hose. Here's how:
https://www.chameleonforums.com/drippers-19143/#post177210
I also think two drips a second is fine. In the morning my drippers are set to drip a few drips a second, which slows down as the water level and hence water pressure goes down. Then slower during the day, fast again in the evening.
And I agree melting ice cubes are unnatural and too cold