Do you get a higher CRI when using CW & Daylight together, seporately or does it change it?  Also does CW light some colors better than Daylight and vise-versa?

The CRI of fluorescent lamps, is depends on the phosphors used and their emission of the spectrum.
Halophosphors have poor CRI. Triphosphors have good CRI of Ra8=85. Deluxe halophosphors (What are used in the US) and Deluxe triphosphors have excellent CRI of Ra8=>90.
Using cool white and daylight fluorescent lamps together, won't improve the CRI, if both fluorescents uses the same phosphors.

Generally speaking no.
For good CRI (90+) you need the light to be emitted in all parts of the spectrum, from deep red till near UV.
The high efficacy lamps do not have CRI above 80 due to the eye sensitivity curve:
The light from blue to near UV and mainly the deep red participate only very little on the lumen output, so the high efficacy lamps are simply designed to not "waste" the radiated power in these spectrum areas.
So the tri-phosphor's CRI80 lamps usually do not emit anything there. The light color is then "tuned" by varying the ratio of the peaks (all of these triphosphor's radiate anyway). So when combining 6000K with 4000K, you get about 5000K, but still about the same CRI.

But for the good CRI (90+) need equal power share as the other colors, therefore such lamps have to spent about 25..30% of radiated total power even in radiating in bands, where it does not add to the lumen output for more than few percent.
And that is the reason, why CRI90+ lamp have about 25% lower power than the CRI80 counterpart of otherwise the same design. T
hey have still the same energy efficiency in converting the electricity into radiated power, but the eye still see it as less light.