We all have our pet ways of doing things, but this is how I'd approach it.
Forget the weed block. If you have invasive stuff surrounding the bed, you can put in the plastic or metal barriers. These are 4", pounded down.
You've mowed. I sprinkle down the cheap bagged manure. Many don't, but I just think it helps break down the grass etc and helps enrich the existing soil. For your 4 x 8, 2 bags. Then layer down with several layers of newspaper, and some use cardboard. I put on another 2 bags manure and 2 bags cheap top soil (mix up and spread out). Water well.
Then build up. Can be same day or next. You already have 4 of the 32 bags needed. Perhaps 20 bags of the good stuff. I like to add a bag of cotton burr (one for whole garden). It's about $10, not at the box store but a private garden center. For bagged manure to mix in I have gone to Black Kow, yellow bag from Lowes. It doesn't clump, has some sand in it, needed for you, and mixes easy. I'd go 2 bags. This can all be mixed in the bed, working wth hands and hand tools. Then see how it is filling in.
I have a couple of 4 x 4 beds that need cleaned and worked. I did just clean out one, turned the soil. After just one year no line between new and old, and there are lots of worms. I have bags of stuff to fill in, enrich, mix with existing. Then plant, and the plant starts are very anxious to get in their new digs! This was all done a month ago. Oh well, not, sigh. Getting around to it! Welcome to the great world of gardening.