Conclusion:
The double barrel could use a slight nerf, because it is noticeably better than your other options while also being dirt cheap. I would suggest reducing the pellet count of the buckshot by 1 or 2 pellets and to increase the reload time. You should need more time to reload this thing than to shoot it.
It's as much the DB and Heavy being good as the other shotguns being bad, and once enemies start wearing decent armor, all the shotguns are bad. And the advantages of shotguns are that they are a little lighter than weapons of the same tier and give an accuracy bonus, but neither of these count for much past early game either.
Ol' Shotgun: 15x8. Even though it shoots "birdshot" it has fewer pellets than a single DB shot.
DB/Sawed off: 20x10. The highest pellet count and the ability to auto.
Domestic: 18x7 reg, 20 stun (ugh), 21x7 chem.
Military/Black OPs: 18x7 reg, 40x1 slug. A 40x1 close range shot isn't impressive. Shares basic ammo with the Domestic but can't use chem or stun rounds.
Heavy: 20x7, 28x3 segmented slug.
Blunderbuss: 24x6, 25x6 chem, 40x1 HE.
Kustom Blunderbuss: 33x4 gold ammo
Combat Shotgun: 22x5, 20x5 inc
Caws: 26x5, 50x1
Arena Flak Cannon: 38x10. Very wide spread means most pellets will miss.
This is a weird improvement curve. Overall damage increases a small amount, if it all, as the shotguns upgrade, and not enough to defeat armor.
Slugs don't do a better job than other guns that become available at the same time. Why would I bother with an unreliable Black Ops or CAWs slug when I can use a light cannon or boarding gun from a safer distance and do more damage? Or use a saber at a closer distance and do more damage?
Compared to SMGs and MGs, the math only favors shotguns if you use them on automatic. The blunderbuss is only an upgrade over the DB if you use them only on autofire. The combat shotgun is worse than other shotguns unless its used on autofire.
Of the currently available shotguns, only the arena flak cannon has a place among my mid-game team, and a big part of that is simply that ranged cutting damage gives some variety. Even so, enemies are usually picked off by my rocket launchers, death blossoms and lasers before the flak cannon can get in range. In a previous aborted playthrough I got lucky with early access to flak cannons, but even then its wide spread and lower accuracy bonus made the DB or Heavy more reliable against unarmored humans.