Thank you so much for the suggestions! I will try them and post my results.
Have you:-
1) Measured the voltage when the supply is turned on.
Yes. I am at 5V.
2) Checked that the ground on your power supply is connected to the ground of the Arduino.
No, the ground from my power supply is not connected to my arduino ground. I had thought of this but was nervous to try,.
3) Have a capacitor across the power supply at the strip power pins.
No, I will try this.
4) Have a series resistor in line with the Arduino data pin.
No I will try this too.
5) Pressed the reset button on the Arduino when everything is powered up.
Yes, I have tried this.
It could be that the power supply is going into current limit and thus the voltage output is clamped. As you turn it off, the current drops and takes the supply out of current limit and restores sufficient voltage to power the strip briefly while the voltage decays.
I have tried two separate power supplies and both do the same thing. I am only testing less then 100 LED's on my longest strip right now so the power supply should be able to handle it no problem.