If you are going to mow the grass short, only do it the width of your rows, leaving the grass between rows (the width of your
lawn mower) makes already covered pathways, keeps more moisture in the soil for your plants to use.
This will also allow you to use your cardboard to simply cover the actual row, or you can simply cut the row grass short and lay on a thick layer of composted manure and plant directly into that.
If you do use the cardboard, you will be punching through it so the plant roots can go deeper into the soil, otherwise they will run across the top of the cardboard (shallow rooting = more
watering).
The way we build our raised beds is one at a time, we have had to move to using
concrete blocks for a permanent
water stop/ diverter setup and to keep the dogs and donkey out of the gardens.
These beds will also keep the
moles from traveling into our raised beds. By only purchasing the blocks for a single 2 foot by 30 foot bed we can afford to put in a long lasting garden space and it will be there forever, it is also easy to sit on the blocks for back breaking work in the garden space.
We use short pieces of 2x8 as moveable benches.
The other good method for making beds in lawn areas is to use string to lay out the individual beds then use a spade to cut through the roots of the sod, roll it up and flip it over and lay the green side down in the trench.
This method works well too, but it is a lot more work to get a bed ready to use this way.
I always try to do a no till/ no dig method when ever possible, it keeps all our microbiome in tact and that means I'm planting in a thriving micro world that will help my plants grow fast and strong.