I've not gotten to buy a 1600SW so this based purely on research but there IS A SOURCE for some parts. Now I'm not a monitor expert, but the fact that your LCD screen itself turns on and then off..to me...screams either backlit bulbs or backlit inverter (or both...they are cheap...so...both).
Both are available as NOS from here:
CCFL bulb assembly:
https://www.niktec.com/sgi-1600sw-ccfl-r...lamps.html
Instructions:
https://s3-us-west-1.amazonaws.com/nikte...NIKTEC.pdf
CCFL Inverter PCB:
https://www.niktec.com/1600sw-lamp.html
They have all LCD and PCB parts...but NO PLASTICS or STAND! LCD is pretty cheap if you've got a scratched one and want to take the time to replace it.
I've seen the instructions that come with most of these parts (I think the seller includes them with part purchase). Replacing the physical backlights is VERY EASY in this design...it was made as a modular/slide-in component (unlike every other monitor in history) that slides into the side of the screen assembly as a long brick when you're inside the frame.
Assuming you've tried the same power supply on both your 1600SW LCDs, I'd assume the above issues, prices are pretty cheap. I have no first-hand knowledge of any of this...only hopeful research of someday getting a few of these 1600SW in great/mint skin condition and refurbing them.