Any way you can cover the grassy areas with something like
cardboard? Then a nice layer of compost/soil, and some mulch on top.
Shortly after you do that, you may want to consider planting something that will offer some cover to reduce the chance of a lot of grasses sprouting again. I personally have not tried cover cropping on my garden area but something like clover seems like a good choice for it's tough, flowers nicely for pollinators, is apparently a nitrogen fixer, etc.
I have found that a thick layer of
wood chips smothers out weeds pretty good and they are surprisingly easy to pull out once they start popping. I have read that there can be some issues w/ woodchips - mainly nitrogen related - but that really depends on age of chips, condition, and thickness of the chips. I got some beans and tomatoes planted in them and they seem to be doing fine though. I do apply a little bit of my
urine every couple weeks to those areas just to make sure since I used fresh chips and my soil quality underneath was pretty poor.