Illustrator color fill problem

You’re doing it right by using rectangles instead of using the dieline to fill the shapes.

Don’t use the shape of the dieline and fill it. Add rectangles and your fills separately from the dieline.

It’s better to keep the dieline ON TOP, on another layer of your filled colors because:

1. Your need to add the bleed and that bleed has to be in 45 degree
   angle in some parts (maybe). Right now you have no bleed on your dieline and it’s wrong.
2. There’s areas of your box where you shouldn’t have fill because
   there will be glue added there
3. You don’t want to print the dieline by accident; this one should be
   in a spot color and set to overprint trapping

This is how it should look like when you’re done. I used some other to make it more visible. The corners sharing a bleed should be at 45 degree.

Don’t put white areas between bleed areas; fill everything with the same color to avoid “stamping” on press (for example, the bottom part is all filled with the same color, the bleed isn’t limited to the edges of the dieline)

And where you know there will be glue, you should leave 0.375″ to 0.625″ of white area there; the glue will stick better this way!

I also suggest you read about rich blacks:

What kind of black should I use when designing for CMYK print?

And bleed:

How can I determine how much bleed to use?

And dieline/diecut:

Creating in PS a bleed for a die cut sticker

Cutout line settings