First you have to understand no camera made has a color accurate screen on it ... not even the $70,000 Red & Sony rigs. They use a heavily calibrated and profiled external monitor on set. And if a $70G camera doesn't, how is your vastly less spendy supposed to do so?
I've been in pro imaging for many years, since the beginning of digital imaging, haven't yet seen a really accurate camera screen. Red, Sony, BlackMagic, Panny ... none.
So ... how do you decide "accurate"?
You have two ways ... first, visually, requires a heavily calibrated system with a signal path out via a breakout device from BlackMagic or AJA, not your GPU ... to that calibrated & profiled monitor. That typically involves a calibration LUT in either the breakout device or the monitor.
Second ... checking the scopes. Especially for neutrals, black & white, the scopes are your most accurate indicator. The Waveform ... my preference is YC no chroma ... is good for tonal distribution and for 'end' black & white points. The Parade RGB is also good for that.
Vectorscope is where you check your saturation levels and for 'clean' neutrals. Put a mask on a clip so you only see a black/gray/white area, and see if your trace in the Vectorscope is in the middle ... no color .. or not.
So for your question I'm not interested that much in your screen ... I want to see what the scopes show. What is your black point showing as? That will tell us and you a lot more about precisely where that image 'sits' fro black or white points than a screen-grab.
And ... if the scopes in Premiere show say a black point of about 3-7, which for most program material is a good area to be (unless you REALLY have a true black) ... then for most things, that's very appropriate.
Past that, what's your monitor and your viewing environment? How bright is the room? What calibration have you done with the monitor, and is it set for Rec.709?