Answer: The straw was placed there to prevent erosion, but if it were 4-5" thick, it would also suppress weed growth. If the straw mulch is not that thick, that's why weeds are popping up. If it is that thick, the weed seeds germinated in the straw and that's where the roots probably are, which should make them easy to pull or to knock down with a hoe. Go after the weeds that are flowering first - you want to stop the flowering/seed producing stage. If they go to seed, you'll end up with dozens, if not hundreds of weeds in the area. I'd knock them down with a hoe and then rake them up. Don't just knock them down and allow them to lay there - they could drop seeds. If you have some persistent weeds, you can use a contact herbicide such as Round-Up, sprayed directly on individual weeds. You'll still want to rake or pull them up after they're dead (so your yard doesn't look like a weed patch). It may be a lot of work now but if you keep the area as weed-free as possible before sodding or sowing, you'll have fewer weeds to deal with after your new lawns are installed.
Best wishes with your landscape!
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