You probably have never seen your strip at full brightness.
60watt of LED light is quite a lot.
A 10watt LED lightbulb can fully light up a small room.
The controller is outputting PWM, just as you're planning to do with the Arduino.
If you don't have a total (all colours) PWM value of more than 128, then average current draw will be less than 1Amp.
No point in upgrading the drivers if you don't have a bigger power supply.
Leo..