My veg garden is a field, the soil is clay. I am learning how to deal with it. Remove turf if possible, well in your case weeds. I am now using old feed bags to surround beds and try to keep some control. Where I did put a good layer of manure last autumn I was pleased to find how busy the worms had been dragging it down and helping to break up the soil. If those beds were mine I'd start by trying to clean up one and feed it well and cover the rest with any material which will kill the weeds. If you can get muck by the trailor load you will not regret it.
Is yours clay as said you have rush, one big mistake I made the first year was digging swimming pools for the veg, no drainage. I now have drainage channels out of the bottom corners of all beds and any trees I plant have deep channels leading downhill, I have to check that I have dug far enough as in some places the run off backs up to the roots and then I dig a longer channel.
Good luck. I am currently digging my new brassica beds, the first trench was pretty good but the one I have nearly finished is mainly clay, I have been removing some and burying the turf(luckily mainly dead due to some good muck) adding lime and then cross my fingers.