There is nothing you can do that will prevent new "weed" growth. If the seeds find a bit of soil, moisture, and the right temperature they will germinate and grow. Keep in mind that if you have a soil which "weeds" will not grow that soil probably will not grow desireable plants either.
Often soils covered with landscape fabrics become water saturated, but over time as dust and other debris from the air accumulates on these fabrics unwanted plants, "weeds", will grow, if there is adequate sunlight and moisture. If you think drainage may be an issue with the soil under your proposed deck correct those problems first.
Turning the current grassy/'weedy" soil under is one choice you have or you could cover that with newspaper/cardboard and mulch but there is no need to do both. The newspaper/cardboard is put down to keep unwanted plant growth from access to the sunlight plants need to grow. Tilliing the soil can bring long dormant "weed" seeds close to the surface so they can germinate and is why, often, we are told to till our gardens, wait a week or two (gives those seeds a chance to germinate) and till again.
If this raised bed is to be more than 6 inches deep it is not likely that much more then just laying that much soil will be needed.

Under the new deck - as you prepare the area, establish good basic drainage. There are a lot of ways to do that, but the easiest is with the final grade. Using a geomembrane/weedblock fabric is a sound idea. If it is well covered (the deck will do a lot of that) then a good quality membrane should last a decade or more. The cover over the membrane should be something that won't give new weed seeds that will find there way into that zone a foothold for germination. Rather than sand, I would suggest stone or gravel for that reason.

The new raised bed -- the soil you are going to move is full of natural weed seeds. Practically, there isn't much you can do about that. It will continue to support germination whenever the conditions are right. As a bed it will also get new seeds from the surrounding environment. Once the bed is built, then a good mulch layer of 2-3 inches will help minimize new weed germination. But like all exposed beds, there will be weeds. For moving the soil to the new raised bed, you can just dig up the old, turn it over and put it in the new bed. Covering it with newspaper or a good mulch will prevent new germination, but when that cover is removed for planting then you face the same old problem. A different strategy, but more work, is to turn over or till the soil after it's in the new bed location, leave it exposed and then cultivate every week or two to disrupt newly germinated seedlings. The object here is to deplete the included weeds seeds by encouraging germination, then destroy the seedlings by cultivation, over and over again, until it's planted, then mulch.