Not driving the gate to saturation means your FET will act something like a voltage-controlled resistor. The more voltage, the less resistance. Don't do that unless you've calculated it to provide a certain resistance, and know it's doing what you think it's doing.
When you want to switch
current, use a FET that has a low Rds(on) (the resistance between drain and source) specified at 5v on the gate. Vgs is NOT the spec you want. Look for something like: Rds(on): 20mOhm @ 5V. <---- you want that number (mOhm) to be close to insignificant. Resistance here means heat.
+5v to your LED's common anode, each cathode to its own current-limiting resistor (sized appropriately for the LED's forward voltage and current rating, but 470R is a reasonable starting point), the other end of that resistor to the drain pin of its own N-channel Logic-Level MOSFET, the source pin to ground. 10K resistor from ground to gate, 220R from gate to Arduino's PWM pin. You'll be solid then.