Will Full Frame solve this?
Probably not. Nothing can make image quality as good as you'll get in bright sunlight when there's less light. But it will help by about one stop in terms of signal-to-noise ratio.
Ultimately if the scene is two or three stops dimmer then you have to open the aperture or slow the shutter time by the same two or three stops to get the same amount of light on your sensor. Using a FF sensor vs. a 1.6X APS-C sensor buys you one of those stops.
How much that will improve the way your images look on the 27" Thunderbolt Display is hard to say. That's a very high resolution monitor with very precise color differentiation that will show the flaws in a lot of images that lessor displays will not.
If you do decide to make a move to full frame I'd strongly urge you to consider the 6D or even the 5D Mark III. The 6D is every bit as good a camera as the 5DII other than the fastest shutter speed of 1/4000 second vs. 1/8000, a flash sync speed of 1/180 vs. 1/200 second, and no user configurable Custom Exposure modes. The 6D also has a slightly better AF system (11 vs. 9 AF points functional to EV-3 vs EV-1.5), a better exposure meter (63 zone dual layer vs. 35 zone), a DiG!C5+ vs. a DiG!C4 processor, wider EC (+/-5 stops vs. +/-2 stops), more flexible AEB (2, 3, 5, or 7 shots at +/-3EV vs. 3 shots at +/-2EV), Multi Shot Noise Reduction, in-camera CA correction, multiple exposure, in-camera HDR, in-camera resizing and RAW processing, a faster max fps (4.5 vs. 3.9), a deeper buffer (1250 JPEG/17 RAW vs. 310 JPEG/13 RAW), and a shorter shutter lag (<60ms vs. 73ms).
The 5DIII is on an entirely different level in terms of the AF system and also has everything the 5DII has and a lot more. Of course it also costs a little more, but having shot extensively with both the 5DII and the 5DIII I can say the 5DIII has slightly better image quality than the 5DII but is a lot more camera in every other way. With the recent introduction of the 5D Mark IV, now is the time to find a deal on a 5DIII, new or used. Unless you manually focus everything the AF system alone is worth twice the price difference.