I think you are getting parasitic powering of the LEDs. I have an LED string that glows dimly even when the power is off. Naturally I quickly disconnected the data lines as well.If the power is off you should also have the data lines LOW.
I don't get your circuit. Is pin 1 of the LED string +5V or Gnd?If it's Gnd, why are you grounding pin 4?If it's +5V why is it connected to the negative side of the battery?
What's the "low"/"high" voltage on the signal that's triggering off the ss relay?
also R1, what value are you using?Maybe it's not allowing enough current for the LED for the fet's to fully switch on?
What current is drawn by the LED string?