Greetings
SoberSunday01, and welcome to the Shotguns forum.
Let's start with your handgun -- what selection process did you follow that led you to the S&W 686? Did you get it on an impulse, or a after considerable deliberation? If the latter, then a similar process can be applied to a shotgun. Let's look a several variables:
Basic action type:
Why did you go with a revolver? Obviously it wasn't for the maximum number of shots. Most likely it was because of a revolver's simplicity of operation… load, pull trigger, bang. Semi-auto pistols vary in operation form model to model; but, modern double action revolvers pretty much function the same. There are some break-open revolvers out there, but most are of the swing-out cylinder design. Assuming you prefer the simplicity of the DA revolver, and that preference applies to you shotgun selection, perhaps you should scratch auto-loading shotguns from your list.
Without the auto loaders in the mix, that leaves standing breech (hinge guns: single-shots, side-by-sides[SxS] and over/unders[O/U]), pump-action, and bolt action guns. For trap shooting, a single shot is adequate. It's the only shotgun sport that doesn't require a quick second shot. Many competition trap guns are basically tricked-out single shot guns. Your trap shooting buddies aren't going to laugh if you show up with a single shot gun like the Browning BT-99.
I mentioned bolt-action shotguns only so you'd be aware of them. Typically they are bottom dollar guns that are very slow cycling. Actually, they a little better than a single shot with an on-board ammo supply. I won't mention them again.
Since you mentioned hunting and HD, you'll probably not want a long barreled single-shot trap gun -- a repeater of some sort is order. For clay target shooting, two shots are sufficient, and many folks consider two shots sufficient for hunting. For more than two shots there are the pump action guns (we already dismissed the auto-loaders). You've been reading this forum for a while, I'm sure you are aware of the complaints posted by the owners of pump guns. Since you may be after simplicity and reliability, a two shot standing breech gun is the obvious choice.
These days, when selecting between a decent SxS and an O/U, most folks will opt for the O/U. There is a group who are fans of the SxS coach guns for HD and cowboy shooting. For trap and hunting a quality O/U will serve you a lifetime. You could opt for a combo -- basically and O/U with and additional long single barrel for trap.
Budget:
Since you have a S&W 686 instead of one of many less costly .357 Mag DAs, I'll assume you're not after a bottom dollar gun. For quality O/Us, Browning and Beretta are the heavy hitters in the lower cost market; however, Winchester is trying to make a come-back with their new line of Belgian made guns. Take some time and familiarize yourself with their offerings. I'm sure others will recommend their favorite guns.
Fit:
When buying a revolver, most folks don't give fit a second thought. If it doesn't work out, modifying or replacing revolver grips is not a big deal. This is not the case with shotguns. It is not unusual to spend considerable effort and $$$ to make a shotgun stock fit you properly. Try as many guns as possible: rent, beg or borrow as many guns, for fit and firing as possible. Only with shooting experience will you learn what stock dimensions work for you. If you're not "average" size, don't be surprised if none of the factory stocks fit you. Thankfully, gun makers are aware that we no longer have a friendly custom stock fitter in every LGS, and they are making guns with self-fitable stocks.
Of course, you may have bought the S & W 686 because it was bright and shiny and you had to have it on the spot. If that's the case, the same process may apply to your shotgun.
Good luck, and keep us updated with your progress.
(Good thing the inlaws have wireless).