When I try to offset my line with dashed stroke, the original and the offset path do not run in sync. The dashes should be side by side, but they are not. Screenshots:
Can someone help?
The quick answer is that you can't. Not with simply applying the dashed line to a stroke. Illustrator doesn't align dashes based on the document grid, it's based on the line and nodes.
If you want them to line up, you will have to manually separate either the outside or the inside line to shift the start point for the dashes.
Billy Kerr's answer below is MUCH better!
You can try also this trick. It's a workaround and has its own problems like other workarounds.
The original curve with a narrow stroke
The stroke width is changed to 3 mm
A copy is made with 3,75 mm wide stroke and dashes
Version 2 is lifted on top and aligned with ver. 3. Ver. 2 is used as an Opacity Mask for 3.
You can edit stroke widths and dashing after releasing the opacity mask.