What are the exact make/model of the GPU and PSU? Also, what game(s) are you playing, and at what resolution(s)/detail level(s) that are causing the screen to go black?
More information (specifics) like this will greatly help in troubleshooting the issue. Details make all the difference in the world when it comes to providing good, guided advice vs. shotgunning or just throwing stuff at the wall and hoping something sticks.
That said, with the information you've provided thus far, your leaning towards the GPU as the likely cause of the problem is a logical place to look. Everything is new except the GPU? Case, MB, CPU, RAM, PSU, HDD/SSD, etc.? I can see why you would be suspicious of the GPU, especially since this was a pre-existing issue before your rebuild. However, I don't think I would jump to the level of specificity you are doing by asking if "it's not drawing enough power". Unless you are wanting to identify a specific problem with the part and (at the micro-component level) wanting to repair that part instead of replacing it (soldering on a new capacitor, etc.) then all that matter is answering the question "which main part is causing my screen to blank when gaming?"
So here's what I'd do: Take the GPU out and do a visual inspection of it. Does the fan spin freely or is there dust in in? Are the bearing starting to seize up causing it to clearly stop spinning faster than it should when you spin it by hand? How do the capacitors look? Are they swollen (tops look like little domes instead of being flat) or have they possibly even blown their tops? Are there any other visible signs of card failure? If you can SEE something wrong with the card then there is a really good chance that replacing the card will resolve the issue.
Here's a swollen/domed capacitor:
And here's an example of a solid capacitor that has "popped" on a video card: