Hello if they're autumn fruiting raspberries they need cutting down to ground level every year they fruit on new wood,summer fruiting need the previous years canes tying in, I know you dont want to use weedkiller but it's the only way to kill bind weed,use round up(glyphosate) at 50 mm to a litre of water in an old jug then paint the leaves with the mix,any coming up in your veg beds use a piece of cardboard and paint the leaves with a brush wear good rubber gloves in a couple of weeks the bind weed should start looking sick ,Glyphosate is a translocated herbicide so needs leaves to work dont remove the top growth ,any run off should break down in the soil jezza