What you're describing is not bad accuracy, but bad sight registration. changing ammo isn't likely to have a great effect on that. (Different weight bullets can affect whether the bullet hits high or low, and heavier bullets typically hit higher (which seems almost non-intuitive.)
If the gun shoots a small group (with the points of impact together), the gun is accurate, but the sights aren't properly registered.
An inaccurate brand of ammo (in your gun) will generally mean a larger "group" of shots, rather than a small one (i.e., shots not clustered together.) Some guns like one brand better than another, but it's not generrally a clear-cut thing.
As suggested, start at a closer distance to get the sights "sort of" right, and then tweak them.
If the sight is physically centered in the rear dovetail, and you can't adjust it enough using the screws, you may have to physically shift the entire sight a bit to the right.
When you can hit the point of aim at 20 feet, then you start moving out. Then you can use the adjustment screws to fine-tune the points of impact, matching them to the point of aim. Set the sights for the distance you'll be using for most of your shooting. Most of mine are zeroed at 30' - 40'; if it's much father than that, in a real-life situation, I'd probably be running and ducking, or looking for a rifle or shotgun.
Be sure to shoot from a rest (maybe a large plastic bag of rice from the grocery store, if you don't have sandbags) and slowly squeeze off each shot. You can't sight in a gun properly just shooting offhand, standing, etc. Doing it with one hand makes it even harder. Rest your hands, not the gun, on the bag of rice/sand/beans, whatever.
