This year you get to pull.

If they've gone to seed, you will have many seedlings to remove as soon as you see them.

The easy way to manage weeds is to avoid seeding. Also to kill them as babies -- small is easy to hoe, to pull, to use a flamer, etc.

Choose your method and use it regularly.

'Pulling' is a slight misnomer for what you need to do. Some weeds, especially perennials, will just break off if simply pulled. You need to get a hand fork and really remove every scrap of root. 4 x 4 is really very small and keeping those two beds clear shouldn't be much work at all. And there's no need to wait for warm weather. This is a task that is ongoing. The hoe and the hand fork are your friends. Weeding is a regular process, like cleaning your teeth. You can't just do it in one big splurge and then hope it is fixed. Beds don't get overrun if continuously dealt with, as Jean said.

And when you harvest all your crops in late summer or fall, you will need to cover the soil to prevent new weed seeds blowing in. The easiest way to do this is to mulch the beds well. Or plant a cover crop that will grow through winter to be turned into the soil in spring, adding both organic matter to the soil as well as nutrients and keeping weeds to a minimum.

Why not lay down some newspaper or cardboard over the "weeds" and kill them where they are? Cover the newspaper or cardboard with a mulch material to hold them in place, and hide them from view, and plant through that when the time comes.

kimmq is kimmsr

Mulching is good either with or without newspaper but if you don't deal with perennial rooted weeds first they'll be back. Stuff like Bind Weed, Couch Grass, Enchanters Nightshade, etc. will just punch through a mulch.

In other words, please post images of what you are trying to eradicate.

A thick layer of cardboard, thick layered newspaper, or black plastic, left in place for a couple of months especially in the growing season will kill most or all weeds. If there is a perennial with big deep established roots, you may have to dig it out though.

4x4 is such a small area, you should be able to dig out any unwanted roots in well under an hour.

Using newspapper or cardboard (usually available free) is a way to kill unwanted plats while using less of the more expensive material (if you need to buy it). To suppress the growth of unwanted plants ("weeds") just with, say, wood chips yo0u would need about a 4 inch depth while the paper could allow just 2 inches to be plunked down.

Keep in mind that those unwanted plants ("weeds") used nutrients from that soil that will need to be replaced if they are pulled out and thrown away. Covering them in place and allowing them to die, from lack of sunshine, will put the nutrients they used back into the soil as well as adding some organic matter that will benefit the soil as well.

kimmq is kimmsr

Pulled weeds are not necessarily 'thrown away'. All mine go on the compost heap. For seed sowing the ground needs to be cultivated so weeding can be incorporated in that process. You can't sow seeds into wood chips, cardboard or the compacted soil beneath them. Presumably the OP wants to get growing asap. Hand cultivation with a fork would allow that.

No organic matter leaves my property either, unless thorny, and bones.

Chickens for a few days will clear it easily :)