I don't have a lot of experience with this, but I would start with making a bulleted list of just the key facts. The ones that support your claim that they violated the rule.
Something like this:
=We bought our tickets via phone on xx/xx/xx for $xxx.xx
=We received our email confirmation with seat assignments on xx/xx/xx
=we checked in for our flight at x:xx xx/xx/xxxx
=we checked our luggage at x:xx xx/xx/xxxx
=we arrived at the gate at x:xx xx/xx/xxxx
=we were denied boarding at x:xx xx/xx/xxxx
=we were finally offered $xxx at x:xx xx/xx/xxxx
You get the idea. Only include in your timeline facts that support a violation. Everything else is drama and relevant for empathy on the internet (which could be good for drawing attention from someone who can help you). You need to be prepared to file a formal complaint, and for that you need just the facts that support your complaint.
I wonder if the fact that your luggage left with the plane without you is a violation of some rule too.