Advice needed on weeding and cover cropping, garden bed maintenance

We're heading into our 4th season with our current home garden and have scaled it up a bit each year. Now we are trying to get better at maintaining what we've started. We get a lot of weedy grasses in our garden and some are a pain to pull. We're trying to get better at cover cropping. Coming here looking for advice, please lay it on me!

I'll post some photos of beds we're working on now (bare with me as I figure out attachments on this forum). We have about 10 beds like this, then 6 rows of planting mounds for melons and 3 sisters style plantings, then a raised bed for herbs and various forest gardening on further zones. It's in the beds like shown below that we struggle with weeds. We broadforked the lawn, flipped its top layer upside down, piled on compost on beds and woodchips (free mulch) on the paths in between. Then we mow the lawn around it. We've tried mulching with hay and with straw but always find weeds in it. We are now adding more compost to these beds and mulch to the paths, and trying to use cover crops more effectively to reduce stubborn weeding (speedily spreading stiltgrass, clumps of I think crabgrass that can be stubborn to pull out).

What cover crops should we be working with and when? I know weeding is part of gardening, but should we be clearing beds at this time of year? The thatch from last fall's dead stiltgrass seems preferable to bare soil until we are ready to plant, basically acting as a mulch. Where we are clearing beds and/or dumping compost on dandelions and things like that, we are hand broadcasting clover seed mix to try and take up the space usefully and be easy to weed as we introduce crops. Then in fall we're trying to hand broadcast winter rye to hold the space, and this spring we are pulling up or crimping the over-wintered rye.

Whenever I refer to compost in this post i mean finished compost blend ready for raised beds. We get a few cubic yards each year for this garden and other areas, some of our own and some from a local supplier.