You bought the second thing first. Light matters more than the camera, and it is not close.
A camera sensor needs light. Given plenty of it, even a modest sensor produces a clean, colourful picture. Given little, every sensor amplifies what it has, and amplification is where the grain and the flat grey come from. That is why your new camera looks the same, it is doing the same amplifying, slightly better.
Your specific symptoms map exactly:
Flat and greyish - a ceiling light directly above you. It lights the top of your head and leaves your face in its own shadow, with no direction at all.
Grainy, not enough light reaching the sensor, so the gain goes up.
Worse for half the day - the window behind you. The camera exposes for the bright background and leaves your face dark, and there is nothing a camera can do about being pointed at a light source.
The order of operations that actually works: light, then background, then camera, then microphone quality. Most people do it in reverse and are disappointed at every step, which is exactly the experience you have had.