The boxes look pretty shallow and tired...
You're going to halt the grass rhizomes and weeds in this process without Nuking your base soil and encourage the base soil to thrive!
Hit the foliage with some straight cider vinegar. Go out there with a spray bottle and spray the grass/weeds. Let it cook for a day or two.
Gather some cardboard/news paper and then layer it on your beds, water it down! Cover this with heavy duty black plastic or (contractor bags split open) and lay it over the beds, staple the plastic to your existing frames with some thin slat or cardboard strips to keep the plastic secured to wood frames.
Let this cook while you gather your new wood and organics to build new beds! Decide how you will water!
Build your boxes! My primary grow boxes are 4'x8'x12". It's a good depth and overall reach and I'm a shortish. Avoid pressure treated wood when ever possible. I'm in High Desert and Cedar is my preference.
I'm also not a fan of rototillers. It slices, it dices and forms a hard pack in your beneficial soil base.
(Uncover your beds. You toss that cardboard/newspaper into your compost bins! Composting is easy or rocket science...)
With a four prong fork, start turning the soil in two passes, across the top of the bed, working backwards, standing on a plank board... you turn two rows and pull out, by hand, as much GRASS rhizomes or weed roots! The plank board keeps you from *compressing the bed!
Free work out and gets your nose into the smell of your soil/dirt!
Turn the soil about two rows, rake and amend as you go backwards, pull out the grass rhizome hair and burn that hard!
Set your new boxes over your beds. Recycle and throw down a good layer of shredded leaves, some brown cardboard, and begin building the soils to an 12" depth. Get PH test stuff. Oddly, some fish kits are cheaper.
Raised beds in place...
Grass is your enemy! Grass should be nowhere near your paths between boxes nor on the outer edges.
A garden is a free workout! A garden is a place of connection!
Happy Gardening!