Here is just a thought. Coming from a compound that probably weighed 4+ pounds to a longbow that probably weighs under a pound can make holding steady and consistently a bit of a challenge. I'm not saying you should ditch the longbow, but you might try adding a bow quiver to your setup, loaded up with as many arrows as it holds less one. The added weight may help you steady your bow arm since you are used to WAY more mass than you are shooting now. Now shoot that one arrow, concentrating on making that ONE shot count. Then go get the arrow and repeat the procedure. Don't take more than a dozen shots in a session for now, but make each shot with that single arrow count. Work on consistency. Make sure you hold your bow arm steady until the arrow strikes the target.
While 48# is not an overly heavy draw weight, you are used to holding probably 15 to 20 pounds with your compound. Taking too many shots in a session leads to fatigue, which leads to poor form, which leads to inconsistent accuracy.
If you can shoot at home, that is a plus. I sometimes just take my bow out and plan to shoot just one shot. I put all my concentration into making that one shot. If you plan to hunt with your bow, chances are you will get just one shot at a game animal. Practicing placing that one single arrow where you want it will pay dividends in the long run.
One more thing: consider taking longer shots than 10 yards, if you have a safe backstop that is. If you are fighting with arrows that are slightly out of tune, 15 to 20 yards may give the fletches time to make the arrows recover and you may find that you are shooting better than you were at 10.
If that is the case, tuning is definitely in order. Personally, I paper tune. I think it makes form less of a factor and just tells me what the arrows are doing. Bare shaft tuning works great if you have consistent form and release, which may not be the case for you.
Hang in there and stick with it. Nothing is more rewarding than taking game with traditional equipment!