My Mom's solution, that also works for me: layers of newspaper or cardboard, with wood chip/mulch on top to keep it from blowing away.
I mean THICK layers of newspaper: 6-10+ sheets thick all over, with at least 1-2 inches of the wood chips on top. Then it still needs a new layer once a year or so in Alaska (where I grew up) and sometimes twice or more yearly in Virginia (where I am now--rots faster in warmer temperatures.) It sometimes works for me to rake off the wood chips, spread more newspaper/cardboard, and put the same wood chips back on. (Raking them around also helps disturb any little seedlings in the top layer, before those seedlings have a chance to get bigger.)
Are you able to pen chickens in the garden? Not while growing crops, but after harvest and before spring planting? Every weed, and especially every seed, that they eat can save you some trouble later!
I've always wanted to make a chicken run all around the garden, like a big doughnut with the garden in the center "hole." That would stop weeds spreading in from outside the garden. (I haven't actually tried it, but maybe someday...)
I have a particular flowerbed that stayed almost weed-free last summer. Every time the lawn got mowed, I put the grass clippings around the plants in that one flowerbed. I was piling the clippings about 2 inches deep each time, then they'd sink down as they dried, then I'd add more the next time. It got thick enough that weeds did not easily sprout through them, so weeding was reduced. Yes, I tucked the clippings carefully around my plants, so the plants could still stick out and get sun. And yes, I pulled any weeds that I noticed while piling the clippings--but it wasn't many.
For removing small numbers of weeds: boiling water works well. Anytime I boil eggs or noodles, I take the hot water outside afterward, and pour on weeds in a few particular areas (mostly the edges of my driveway.) It might work in your garden aisles, as well. Of course, it kills any plant equally, so don't use it too close to your crops, but it does not leave any poisonous residue behind. And the sight of a particular weeds lying there cooked and dead makes ME feel a lot happier!