Turning Sod into garden beds, how?

I really like the idea of planting into my ground rather than making raised beds. Last year I scattered all kinds of seeds and mulched, but few plants could out-compete the grass. I also have patches of Bermuda grass, which make things frustrating.

Yesterday I got a wild hair up my ass and decided to make a mound for squash. I dug up the sod with a little dirt as possible (still a lot of dirt) and piled it up enough for one zucchini plant. I noticed last year that when he got heavy rainfall, all my zucchinis on ground level rotted. Then in the vacant space I added course compost (the only kind I seem capable of making) and covered it all with a few layers of newspaper, leaves, and straw.

It was a bit of work, but not unbearable. I still feel this is too similar to tilling. I only plan on doing this to get a bed established as a garden, so it is really that bad in the long run? I have a few more beds I want to put in and I am going to just over turn the sod and lay it grass-side down. I love sheet mulching, but I simply don't have the time or means to scrap together that much organic material. I would also love to just lay down some news paper/cardboard and add compost on top, but I don't have that much compost either.

Any help here?

In my opinion it is perfectly ok to dig a garden once to establish it, especially if it means getting rid of a chronic problem like grass. If you don't have a source for compost or mulch, you'll need to have some area of the garden to grow compost/mulch materials, which can also be food plants. A good cold season plant to grow for mulch is Fava Bean. Holland Greens is also a good one.

https://www.superseeds.com/collections/greens/products/holland-greens-tyfon-42-days

Boatloads of compost. That's the only long term success I've had without constant vigilant attention (which pretty much means it's the only long term success I've had).

Turning sod over by just cutting the sod loose from the soil underneath is not a bad thing, it will kill the grass plants and that keeps them from coming back.
Many people "double dig" a new garden space and when you do this and add in the usually large quantity of compost, you are in essence building the soil microbiology as well as loosening the soil well.

Cardboard and newspaper under mulch is going to do the same things but at a very slow rate, many people need those garden beds for growing now not next year.

Raised beds are good for areas that can get soggy. They are also good for spaces where you don't want to disturb other food plants roots (think orchard or vineyard)
Squash and melons do better when they are raised on either mounds or are planted in raised beds.

Raised beds don't need to be created out of soil, as an example, we use straw bales for our raised beds and grow broccoli, tomatoes, squashes, strawberries, peppers, Brussels sprouts and beans in these straw bales.
Once the bales have begun to deteriorate (after one good year of growing usually) we use the now partial composted straw in regular beds as a top dressing or mulch we also use this around all our orchard trees (apple, pear, plum, peach, mulberry, fig, etc.).

Think of your particular conditions and needs first, then use the best methods for each plot of garden. This may not be the same for every garden spot.
On Asnikiye Heca, we have many different areas of gardens, no two are alike in makeup (except for lots of rocks), this means each garden plot has to be tested and remediated individually for best improvement of the soil.
When I test our soil I look at lots of items; soil makeup (jar test for stone, sand, clay, humus), main nutrient profile (P,K,N, Mg, Mn), micro nutrient profile, mycelium content, bacteria presence and typing.
This gives us a complete view of that patch of land's needs.
from that information we can make sure we are improving the microbiology of the soil and not over nutrientizing the soil, something that "normal" soil testing can lead to with out meaning to,

I can tell you how I've done it. But it relies heavily on the fact that I stalk the local tree trimmers. Whenever I see a arborist with a chipper I stop and ask if they'd be willing to dump their load of chips at a house around the corner.

I either put down a 12 inch thick layer of chips and spend a year watching for the highly agressive native grass to put up runners (I've seen them grow through the middle of a ten foot pile of mulch) or I put down a double layer of cardboard and just a cosmetic layer of chips on top.

For the first year I plant by digging pits in the much or cutting holes in the cardboard and putting my plants in the soil underneath. In one case I did sweet potatoes as the first plant and by the end of the season I had wonderful tilth in what had been rock hard soil. For the other beds it's a slow but steady progression to healthier soil.