How can I have two overlapping lines without it appearing darker?

Are you seeing this issue...

• on screen
• in print
• both

What is the color of those lines? How are the aligned to the object? Inside, outside or centered?

I'm suspicious the alignment is other than centered, which would cause the stroke weight to double along common edges.

1. Create an Object Style with NO STROKE (0pt)
2. Draw those rectangles with that Object Style.
3. At the end change the Object style to the stroke width you want at the end and use the center line to stroke option.

Now are the lines exactly overlapping.

Make a four-cell table, and set the cell strokes to the weight you want.