I wouldn’t jump to testing injectors yet.
Have you removed the air plenum and inspected it?
Is there an intercooler on this?
My guess about the really dark while engine cold then going grey is less fuel because the engine is warming up so the cold temp circuit is working so as it warms up less fuel is getting pushed through.
Lack of rpm acceleration and the excessive smoke makes me think lack of air getting to the engine.
Posting pics is often helpful, we might spot something you don’t notice.
Posting videos to YouTube or similar then a link to it here is VERY helpful.
Aeration of the fuel is a passibility. Get a clear line on the fuel return line of the ip (injection pump) in place of the factory one. It is 1/4” diameter about 5 or 6” long coming out the front top section of the ip. Fast & easy is hardware stores sell some that is not rated for diesel but will work and can last maybe 4 years. Best is from fuel-line.com as theirs is rated for all fuel. Test runs so far is 5 years and counting. Once installed just leave it there permanently- this is test#1 from GM when diagnosing anything fuel related. Watch the line with it idling and it should be clear & clean fuel running through. Any trail of bubbles or other contamination is what you are watching for.
I will post a video of the worst case I ever seen where the truck would still run. This is a 1988 with a 6.2 so the engine looks slightly different than yours but is same basic engine. His was so bad he had to start it with his foot on the floor and it barely got to 1000 rpm. When he tried to let out the clutch the engine would die.
After doing this test with him, he replaced the fuel line that was between fuel tank and LP (Lift Pump) which was sucking air in. Completely solved the problem.
If you find the need to replace your rubber fuel line- sae30r9 or greater. All our modern diesel fuel has ethanol in it so it has to be rated to withstand it. The original rubber fuel lines do not and deteriorates from the inside out.
I have seen less air intrusion keep 6.2 and 6.5 from even starting.