That is a good use of that heavy cardboard. I use cardboard from those boxes and lay them down on the pathways of my garden, and then cover the cardboard with grass clippings. Looks like a green carpet. The cardboard breaks down over the summer, but it serves well enough to act as a weed barrier for that growing season.
Sounds like you are doing better than us! To be fair to myself, I am talking about one 13 gallon kitchen trash bag per week, not the larger trash bags for the cans out in the garage which I think are maybe 33 gallons.
I live in God's country. We have to haul our trash to the landfill/recycle center ourselves. No garbage pickup for me. I think that helps me think more about how much garbage I am willing to load up and bring to the dump. Recycling products for reuse at home saves me lots of time and labor in how often I have to make a garbage run - which for me is a 30 mile round trip task.
We do have few recycle bins on the way to town, and Dear Wife will drop off a bag or two of plastic, metal, glass and some paper products to those bins every week. We take advantage of the recycle bins. But non-recyclable garbage needs to go to the 30 miles round trip to the main dumping station.

Dear Wife is a treasure, but I still have a challenge to get her to give me all the cardboard products she routinely uses up. She is more into cleaning up the house than reusing our paper products for the chickens. So, for example, an empty tissue box gets tossed into the paper recycle. An empty cardboard paper roll from paper towels in the kitchen or empty toilet paper rolls from the bathroom get tossed into the recycle bin. Her junk mail gets thrown into our recycle bin. Empty cardboard food boxes get tossed into our recycle bin. I will take out all those items and remind her that I can shred up all those paper products for use with the chickens.

But, it's like talking to a wall and the next day I will find more of the same stuff in our recycle bin. Sometimes I can reduce our recycle bin output by as much as 1/3 to 1/2 of the volume just by shredding up those products she knows I could shred at home. And sometimes she gets mad at me for "dumpster" diving in our recycle bin at home!? Any advice in talking to Dear Wife about this issue? 33+ years together and I cannot get her to change on some stuff....