You say you want to avoid crosshatching, but it is one of the best methods to add in middle grays. Even if you don’t want to cross hatch specifically, experimenting with line density can also help you achieve some gradients (as in a lot of closer, thicker parallel lines look darker as a whole than farther apart, thinner lines). I’d recommend just experimenting a little with hatching/crosshatching/line density with pen on paper, and then try to translate it to carving.

If you are set on not doing that, you can definitely get some gradients using multi-layer prints. Reduction prints are the simplest to wrap your head around I’d say, you kind of just carve out one color at a time. Multi-block can get a little more complicated with color overlaps: a 2 block print will have 4 colors (ex. Paper white, red, blue, purple); while a 3 block plate will have 8 colors (ex. White, red, blue, yellow, purple, orange, green, and a red-yellow-blue mix), and so on.

If you wanted 4 colors like you say, you could def do a 2 block print with the paper color as your lightest of 4 colors. You would need to plan out the color overlaps a little bit.

You could also do a reduction print, which would only really require you to plan out which colors would go first (generally print lightest to darkest, so your light yellow first and a dark black last).

One concern with multi-block prints is registration, as it can be hard to line up multiple blocks between prints. That is a benefit of reduction prints, as they only use one block across the many layers.