Your entire ram lamb does need to be taken off the dam by 4 1/2 months, but ewe lambs can stay until tupping - the dams will wean them themselves by about 5 or 6 months. This allows the ewes to dry up gradually so no need to put them on poor grazing, and instead they can put on condition before tupping time, on good grass. The difficulty is the entire ram lamb - next time, castrate before 7 days with a rubber ring - saves an awful lot of bother later. At what age are lambs of your breed sent for slaughter? I would think that until then he would be ok in with goats, as long as he has no access to their hard feed, which I believe is unsuitable for male sheep.
When we have had to wean all lambs, we have found that the easiest way is not the recommended 'take the lambs as far from the dams as you can on your land' method. This causes stress to both ewes and lambs, as sheep can hear for long distances, so shout to each other for days. Instead we put the lambs in an adjoining field, with just a wire stock fence dividing them. That way, they can still be close to their dams, and even cosy up together through the fence, but they can't suckle. As by that age lambs are no longer hanging around their mothers anyway, this method works well - no shouting and stress at all. For our tup lambs coming off at 4 1/2 months, they have to go across the road anyway to be in with the stock tups, and they do call for a few hours, but not for days as ewe lambs would.