Depends on what they are. Different approach for bitter cress than for sheep sorrel or oxalis.

Those must be large raised beds if you are considering tilling--or you have one of those mini-tillers.

Tillers will just churn the soil and leave a large percentage of the weeds at the surface.

How much headroom do you have in those beds for adding soil?

Better to wait a month and turn over the soil deeply with a shovel--sooner if the bitter cress is already spitting seeds.

The shovel method leaves the weeds deeply buried.

By whatever organic method, you will be fighting weeds for some time in these neglected plots.

Weeds are a ongoing chore. I do not use insecticides or herbicides or any other sprays in my garden. For a few years I put landscape cloth on my raised beds in the fall after all was harvested. It worked great to keep the weeds down. The last couple years I was too lazy about it and did not use the cloth and the weeds are coming back. I have to be diligent to pull them up right away before they set seed. Also, it is not only the seeds already in the soil but, the seeds that are carried by wind and birds.

What bboy said. Depends entirely on what weeds you have. Small annual weeds, like shotweed (what he's calling bitter cress), if they have not set seed, can be tilled in. Things that run, like sheep sorrel and oxalis, or have deep roots, like dandelions and cats ears, cannot be tilled in without coming right back or even spreading.

Winter cover crops help keep weeds down, in the future. I also mulch with cardboard and redo it every year.

I don't see any call for replacing the soil.

For covering the small annual weeds --

Spread an overlapping layer of of a few newspapers & add weed free compost as mulch. Plant transplants using a trowel to pierce through the newspaper. Should be easy with the wet spring weather at work.

Seeds can be sown on top of the compost.

Then keep up with the little weeds that may sprout from blown in weeds from surrounding area. Mulch again after plants are sprouted & soil has warmed. If the weed seeds don't have light they don't germinate, so if needed keep spreading layers of newspaper, cardboard, etc. over paths & around plants -- not touching stems.

This worked for us a few years ago when we had a lot of the bittercress. Beware of that stuff as it blooms in 6 weeks sending seed everywhere when you pull. I think that plant also resprouts from portions of the stem or leaves, so remove it from the garden