I have overhauled probably hundreds of coaster brake hubs in my days; here are a few tips on overhauling/adjusting them.
Loose balls are very difficult in a coaster brake hub as the cup that the bearings sit in is very different from a typical hub, headset or bottom bracket. I highly recommend spending the whole dollar and replacing the retainer as well. Pack the bearing with grease until the grease is coming out the other side, then add more after the bearing is sitting in the cup. Don't put the bearing in backwards; that'll wreck the new bearing in a hurry.
Always do the hub adjustment from the drive side (with the cog). Leave the side with the brake arm alone (locknut tight against the brake arm). Even during the overhaul, there is no reason to loosen that locknut.
As with any hub adjustment, tighten the cone/locknut on the drive side only enough to take the play out of the axle. It's trial and error; 1/8 turn can be the difference between too loose and too tight.
You can almost never have too much grease in a coaster brake hub, kind of like you almost can't overcook a braut. The photo above looks to me like the hub was dirty and not enough grease.
Good Luck, Steve