Baby roots are delicate and need to come into contact with good soil immediately- IMO they haven't the strength to go through mulch or cardboard. This would vary by crop but by and large I would say no.
The time to do the cardboard and mulch process is in the fall.

Seems you've bitten off more than you can chew space wise- maybe scale down a bit and do only one bed the right way with properly amended soil?
I know it is not what you want to hear, but starting plants off on the wrong foot is setting yourself up for failure.

Starting a new garden can be frustrating! I feel your pain. I might recommend starting the seeds in trays or pots, then transplanting into the ground when they're big enough to handle. That might help you deal with what you've got this year.

I wrote this for someone else, actually 2 other people, but hope you may find some value in it also...

If you've got more time than energy, like I do, smothering and lasagna is the easiest way to start a new garden bed for free, or almost. Sooo much easier than digging up grass. Just spread newspaper (about 10 sheets thick) or cardboard, overlapping well, until the area you want to be a bed is covered. Then cover the paper with 4-6" of finely shredded mulch and wait for the grass to die, usually 4-6 weeks but could be longer for some grasses. I've done this many, many times.

My latest one is really ugly but I'm just trying to make lemonade out of lemons with this one... drought (probably aided by grubs) killed the grass here so I decided that would be the new sunny front bed I was considering. I did dig out a little spot that had hardly any grass and put some Cannas and Gladiolus there, a tiny baby maple tree, some Hibiscus cuttings which still just look stupid 'cuz they're "dead" sticks in the ground, then kind of working around it with smothering, and bark chips, which aren't my preference but I had them available. They don't stay in place if it ever rains really hard. Anyway, with this, I'm not planning to leave the bark chips there, they're just making sure the newspaper is held firmly to the contours of the ground to block the air and light from reaching the grass, which is what is needed to kill it. Whenever I can find more shredded hardwood, I'll replace the bark chips.

Anyway, the newspaper decomposes and does not need to be removed later, just dig through it to add plants in the ground.

I've also smothered grass with stuff that was handy, but does have to be removed to use the bed, like sheets of metal, old egg crate mattress topper, the bags of mulch that will cover the spot, whatever's handy. I think it's easier to wait for the grass to die than dig it up, and I don't mind if it has to get more ugly in the process of getting more pretty.

One other benefit of smothering with a leave-in-place substance like paper or cardboard is that the weed seeds that may be in the ground are unable to germinate as they might be if you just dug up the grass and/or tilled.

The lasagna comes into play if you add amendment layers to your smothering. For example, you could put the paper/cardboard, then kitchen scraps, ready to use compost, leaves, yard trimmings, whatever organic material (OM) that is handy, then the mulch (or not, if the other stuff is a thick enough layer to hold the paper in place and block the light.) It's not necessary to have lasagna layers when smothering, but when planting later, there's a huge improvement if a lot of
OM was placed there.

AND, while...

Thank you for your input. I had a 10 by 20 space last year, which was tilled with a small tiller, it had three different beans, and some summer squash, a 6 by 6 with tomatoes. All did very well, til the squash bugs...But I got quite a green bean harvest and had plenty of tomatoes. That success prompted me to expand. having it all plowed was a bad idea.

I am going to pick out areas where the soil is workable, and cover and mulch the rest. The weeds and grass are loving the rain we have been having.
I have three large bins composting...We have chickens, turkeys, and rabbits. I expect by fall to have enough compost to amend/build up those areas of soil that are hardening now. Does this seem like a workable plan?

Absolutely. I was thinking about this yesterday when I was watering very small newly-planted areas on either side of the front porch steps, where I did bother to dig out grass last month, sifting through well to make sure I got all of the roots. (Which took HOURS, btw.) There was never a problem with drainage there until I disturbed the ground like that. It's right where the runoff from the roof hits, so I definitely notice the difference between how any amount of water less than a monsoon type rain would just soak right in there. I had to stop a few times at first or the not-very-strong shower from the hose would have runoff into the edge. (The thought of watering grass next to a border where I have to trim it anyway is horrifying, but I digress! LOL!) Disturbing the layers and tiny tunnels the soil-dwellers had dug ruined the drainage.

The key to fixing it is to pile on the organic matter so the critters come back and fix it AND so it is covered and does not bake dry so easily/often. If you're dedicating areas to recuperation, a lot of material could be composted in place, which eliminates a lot of heavy work of moving it later.

When I have large quantities of something, I try to spread it around to its' first and only destination instead of putting it in the compost. I'm kind of a composting wack-o but I don't like moving compost or any hard work that's not absolutely necessary, I admit.

Also, really heavy stuff that's mostly water, like melon rinds, I see no reason to carry so far. In the shade, they'll melt in a few days, in the sun, they'll be baked dry by the next day. Put near the base of some thirsty plant, the moisture/mulching will help a little.

Compost doesn't have to happen in a bin. I've got to the point where I only compost stuff I don't want to lay on the ground somewhere, for whatever reason. Also, I'm giving up on the dedicated composting spot idea. I'm just going to have alternating veggie plots. One year spot a is for growing and spot b is for the compost pile. Next year, switch, alternating each year. Then I can remove whatever compost I want to other areas, but never have to worry about the bulk operation of loading and moving heavy loads of compost to the veggie plot ever, ever again. After a few years, when the soil is awesome in both places, it might shift to a constant small pile in the corner of each bed. It sounds like you may have enough space for this.

If I really need more space for composting, I'll stick to using 55-gal. drums and/or baby pool (with a hole in the bottom.) So much less loss, semi-portable, and DUMP-able.