By leaving the sod and installing cardboard, you are creating an non penetrable layer beneath these plant beds. This impedes drainage and soil organism activity. I would get rid of the sod for sure, forget the cardboard. Just get soil in the boxes over the double dug beds. Your beds seem large enough to allow them the garden soil and community.
There are lots of gardening trends being thrown out to the public right now, so be careful. They do not work miracles they do not benefit our gardens and harvests.
4X6 is a nice size for a plant bed. Use the garden soil but first double dig the soil under the lawn after removing the sod and turning over in your compost pile. Start at one end and remove a good foot of soil, dump into a wheel barrel or a tarp. Keep going down the bed. Digging and turning over the soil. Chopping up big clumps. When you get to the other end drag the soil in the tarp or the wheel barrel to fill in the last bit of digging. Let that soil build up high. You will rake and compact it and it will fit in your constructed structure.
Do not worry about the grass growing or even weeds when buried under more than 2 inches of soil. No cardboard is necessary nor beneficial.
Fill your bed with soil, compact with a chunk of plywood, then plant. After the plants are growing (about the 3rd set of leaves) you need to add fertilizer, the formula depends on what plants you have decided to plant and how far along your baby plants have grown. Use a balanced fertilizer with N P and K. Use HALF of the prescribed amount. Where fertilizer is concerned, Less is Best, More is Death and None is Dumb.
Decomposed compost, DECOMPOSED ONLY compost should be spread on the bare spots once your plants have 3 sets of leaves. This feeds the soil organisms, NOT the plants. The Decomposed organic matter also helps to inhibit the growth of weeds. This compost helps to hold the moisture in the soil longer. The soil organisms come up to eat this stuff then dive back into the soil, pooping it out in your soil bed, mixing this organic matter into your soil better than you would be able to.
Compost is not fertilizer nor is it a soil replacement.
Once your plants are growing vigorously, water infrequently, allowing the soil to dry out before watering. Depends on your soil, your rainfall. Do not allow that soil to stay wet. That means the roots aren't getting enough O2. Root rot and all of that happens quickly.