If you can hit flying targets and birds with a shotgun with just a bead or two beads on the rib or barrel you don't need a different shotgun to shoot coyotes.
Most of my shots with a shotgun on called in coyotes is way more like waterfowl hunting or dove hunting than it is like turkey hunting.
But, maybe your style of hunting plays a role in that?
For example, using rifle tactics on a shotgun stand will likely result in still shots. The shooter makes minor adjustments to the shotgun POA as the coyote approaches and whistles to stop or waits for a pause. The shotgun is already aimed at the coyote and little/no movement is required to shoot.
But if the shooter lays on his back and suddenly pops up like a goose blind to surprise the coyote up close, you're gonna get a running shot.
Terrain obviously effects your options.
I use my 12g more like a rifle than a clay/skeet gun but that's surely influenced by my tactics and terrain.