I'm no expert, but from what I have read and seen, here are two ideas:
1. Cut the sod layer and just turn it over. Yes, that opens it up a little to oxygen (tilling), but you can do it in largish sheets, which minimizes it.
2.
Cardboard & newspaper sheet mulch. Cover your grass in soggy cellulose layers. You can punch through and plant shrubs in places, but just covering the grass and blocking sunlight will help.
After you've done the above, you can add other sheet-mulching layers atop the cardboard and newsprint, and sow your mowable meadow mix, if you'd like. Doing a season of nitrogen fixers and nutrient accumulators to mulch-in-place, before you go ahead with your target crops isn't a bad idea. Heal the soil first by adding diversity and pioneer plant species that will heal the soil. To paraphrase
Geoff Lawton: focus on fixing the soil biology first, and the soil chemistry will take care of itself.