HELP! Sticker (bur) weeds have taken over my back yard!! I also need to regrow gr over the bur patches?

My (huge) backyard has been overtaken with bur/sticker weeds. I have two dogs that come in every day covered in them. We bought weed killer and a spreader but I'm also concerned about the gr regrowth. Since the entire yard is covered in these freakin things (in some areas they are the only thing there and they have taken over so that gr couldn't survive). I'm worried that the weeds will all die and then I'll just be left with millions of dead sticker weeds and no gr to cover them back up, so my poor dogs don't have to suffer being covered in them (seriously, my shorthaired lab came in with probably 100 of them on her back and legs). Is there a way to kill the weeds and also reseed my lawn in the patches where the weeds are particularly dense? I don't know anything about this kind of stuff so advice is greatly appreciated!