Wet soil is in my experience the best aid for pulling weeds, roots and all. The answer to controlling them in the future is not weed killer (after all, this is toxic to all plants, except for Monsanto's GM crops), but rather mulch. If you cover all bare soil with a thick layer of mulch the seeds of those annual weeds have no chance, and your shrubs and trees will be the happier for it as well. I mulch often with barley straw because where I live it is plentiful, cheap, and easy to transport, store, and apply. It really controls the weeds while maintaining moisture and adding organic material. You might find that rice straw from the central valley is similarly useful. But if your garden is a more formal space (mine is strictly for growing food), then your choice of mulch will probably depend on aesthetic qualities as well as availability. Some people use a layer of cheaper mulch like straw or leaves (or of course breathable landscape fabric) overtopped with something more decorative, such as bark, wood chips, pecan shells, stone, etc...

More than likely the new "weed" growth within two weeks is from new seeds and not regrowth from the previous plants, so you either have a lot of these seeds already in your soil or there are plants somewhere producing seeds that are being deposited in that bed. Controlling "weed" growth in a bed with a ground cover growing in it can be difficult because any plant poisons will also kill the ground cover and placing newspaper or cardboard is very difficult, as is any other type of mulch. There is a bed of Ivy that had a problem with Nightshade growing in it that I finally did pull all of the Nightshade roots from several years ago and have since kept new growth of Nightshade pulled, but this is an ongoing job since birds eat he berries Nightshade produce and drop the seeds in those berries which then grow into new plants. This may well be one of those jobs where you need to spend and hour or two a month doing.

Even a mulch may not stop new "weeds" from seeds deposited by birds from growing, but a good mulch that keeps any "weed" seeds in the soil from getting access to sunlight will help keep those from growing. 2 inches is a bare minimum and if I can only put that amount on I will cover the soil first with newspaper as a "weed" block. Some things, such as Quack Grass, will grow through any mulch you put down unless you first dig out all the rhizomes in the soil and put down a barrier that will keep new rhizomes from invading.

I grow organic and have done so for 30+ years. Never used a synthetic chemical on my property. Never bought a bottle of RoundUp. I am a purist, for sure and I'm very happy as a grower with the way I garden and the results. I use mulch extensively to control weeds. I have used some landscape fabric under pathways. Around shrubs and trees, we use chopped leaves over cardboard or bark mulch over cardboard. We save cardboard boxes and cut them up with a box cutter in the spring and lay them down and cover with either the chopped leaves or bark mulch depending on the application. And if you don't have enough of your own cardboard boxes, if you have Freecycle in your area, someone is always posting they have free boxes to offer. We have a lot of trees in neighboring yards, mostly Maples and one of our neighbors rakes his leaves to our lot line and pushes them over into our yard, by mutual agreement. So we have a lot of leaves. Some are Silver Maple leaves which are small and break down fast, and don't really need to be chopped. Others that are larger, we will just lay out on the lawn on a dry day and mow as usual. I love this mix of grass clippings and chopped leaves as a mulch in my vegetable beds. In shrub beds, a layer of cardboard in the spring, covered by a couple of inches of bark mulch will last all season without weeding for the most part. We try not to replace bark mulch every year. Sometimes if we put it on thick enough we don't have to replace more than every other year. And we can feel great about recycling all that cardboard. Better still, earthworms love cardboard. And we don't use bark mulch across the entire shrub bed. The shrub bed is along the lot line and a distance away from the areas we use the most, and some of it is under the dripline of trees, so we use chopped leaves on most of the bed and just use bark mulch along about a two foot strip along the front of the bed. That works for us, and I'm happy with it. It cuts down on the amount of bark mulch we have to use. In my most active bed, a 20x20 combination of perennials, shrubs and annuals, I mulch over the winter with chopped leaves, remove the mulch in early spring to allow for reseeding and to get a good look at what is coming up. I keep a barrel of the leaves on hold in the garage until I've finished working in the bed. When I've moved what I want to move, sown seed where I want to sow it and then whatever bare ground is there, I will add some chopped leaves as mulch. But really there's not much bare ground left. [g] I don't use cardboard in that bed, I don't seem to need it. As for Preen - never considered using it and because of my basic decisions to avoid synthetic chemical use, I don't end up having to think about what is in it, or what it does. I can just ignore it and garden. [g] But for your benefit here is a link that discusses the ingredients in Preen... Ingredients in Preen