Once I figure what my sights are doing in relation to where the bullets are hitting, I do what I can to regualte them. After that, I rarely shoot at paper. Since you are interested in small game, walk your fields and use pop cans, trash, etc for targets and shoot from field positions. If there is a rest handy, use it. If not, kneeling and sitting make good solid positions.
I can easily take small game and rodents with my Pocket Police revolvers and have done so to 40 or so yards. I have taken targets at much longer ranges. Both are a lot of fun.
If you do get one with a short barrel and rammer, use a 6"length of pipe to exted the leverage. Slip it over the rammer and you'll be pleased with the extra length.
New front sights are not hard to dovetail and filing the rear notch is doable. No reason to live with a revolver that shoots high, low, left or right.