In my experience (and hopefully others can add to this) shooting left or right of centre with a shotgun is caused by one or more of 3 things. First is placement of the butt on the shoulder. Many individuals tend to place the stock on the ball of the shoulder, instead of in the pocket, causing the shot to pull to the left for a right handed shooter. Practicing the mount is (as I keep harping) very important to consistent shooting. Second, the offset of the stock, known as cast, can be responsible. A problem with cast is usually the result of too much cast, not too little. Many shotguns (such as the Ruger Red Label) have no cast; that is the stock is in a direct line with the barrels. I find that this causes me to tilt my head in an unnatural manner, and that is hard to do consistently. For a right handed shooter, I believe that the stock should be cast off (bent towards the right hand side of the gun) by about 1/8th of an inch at the heel, maybe as much as 1/4 inch at the toe, for an individual of average build. A left handed individual needs cast on (bent to the left). Cast is difficult to adjust and can be done by sanding, or better yet bending the stock. In either case, it should be done by someone who is an experienced stockmaker. The measurement changes are small, and it is easy to screw things up. Some shotguns (such as the newer Beretta gas guns) come with replaceable shims which change drop and cast, and are a very good way to achieve a close fit. The third cause of left or right shot placement is poorly regulated barrels, including misaligned ribs and chokes. Rarer in these days of CNC machines, but not unheard of. Unless you can solder barrels or ribs, there is nothing you can do about it. Some gunsmiths can "bend" barrels (don't try this at home), and facilities like Brileys' can manufacture eccentric chokes to correct the problem (it had better be an expensive gun to make this worthwhile).
Adjusting for point of impact by moving your feet might work on the skeet field, but you would have to have a very good memory. It will not work in the hunting field, where you never know where your feet will be when you shoot. V7, with your handgun you are adjusting point of aim by training your body. With the shotgun we want to change point of aim by adjusting the stock. Think if you could twist and cant the grip of your pistol, so that when you opened your eyes you would be looking straight down the sights. With the shotgun, you want to ignore the sights and have the gun shoot where you look.
The good news is that finding your point of impact significantly to the left or right of your point of aim is quite rare with a shotgun. It is fairly obvious if you are looking across the barrel(s), rather than down along them. Also, the pattern from a shotgun will easily cover a 3 or 4 inch difference between point of aim and point of impact. A stock with a little bit of cast to allow for a natural placement of the head is usually all that is needed to ensure a straight shot. You cannot even begin to fine tune for cast until your mount is truly consistent. Far more common is shooting high or low. People do miss targets by shooting behind them, or less often in front of them, but this is because they have not yet learned the proper lead, not because of problems with gun fit. Work on your mount, do the tedious work on the patterning board, get the gun shooting where you look, and you will find that you are a better shot with a shotgun than you ever thought you could be. Finding a good coach or a gunfitter will help speed up this process, and there are many books written on the subject which will help, but there is simply no substitute for the grunt work (what a surprise).
Now comes my philosophical question - why would you do all this work to achieve a good fit, on a shotgun with poor handling and balance, or with improperly regulated barrels or chokes, poor triggers or a questionable bolting system? Gun fit is one thing, handling and balance are another, and mechanical integrity and function are yet another. A good shotgun, well fitted, will last you a lifetime, and be a source of pride and satisfaction. This does not mean you need a Purdey or a Holland. A good many people have found life - long satisfaction with a $250 Winchester model 12. For me, balance and handling are of primary importance. Short of having a moment of inertia table to measure such things, the best way to determine handling qualities is by feel. Pick up and shoulder and swing as many shotguns as you can possibly lay your hands on. You will feel the difference between them. The best clue is if a gun feels lighter than it is - if you guess a gun weighs 6 3/4 lbs., and it really weighs 7 1/2, then the manufacturer has probably got the balance right.
A shotgun is a sophisticated tool, and being a good wingshot is very rewarding and satisfying. The sport is, IMHO, well worth the effort you put into it. I wish I could change the "Oh, so Macho!!!!!" subheading of this forum. It implies mere brute force, and is not inviting to the female shotgunners out there (Canada has some world class female shotgunners). Plus, it reminds me of the Village People (something I can do without). Perhaps "The power and the glory" would be a better subheading. Surely someone can think up a better subheading than that.
Sharptail
