Strategy:Shotgun Types
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General
Shotguns work slightly differently to most weapons. They do not have a projectile - instead they attack a cone-shaped area in the firing direction.
Shotguns can't miss unless the target is not actually within the cone. Instead, their penalty at long range is damage drop-off, which makes them deal a little bit less damage for every tile between the shotgun and the target. Each shotgun has a different damage drop-off, with the combat shotgun being rather good at long range and the double shotgun being completely awful at long range.
Shotguns will make up the cornerstone of the early game firepower, and remain useful for a long time even if focusing on another weapon type due to their perfect accuracy and good, efficient damage without desperately needing shotgun-specific traits like most weapons do. Shotguns excel at attacking groups of enemies at close to medium range, as each enemy will take damage, even if they're behind another enemy.
Shotguns are particularly good at two important tactics - Corner-shooting and Radar-shooting. Radar-shooting means firing shells down a corridor or in to a far away corner of a seemingly-empty room. Enemies cry out in pain when they take damage, so if you hear an enemy yell after your shot, they're hanging around just outside of vision range. Shotguns are good at this because they don't miss (although, they do have a maximum range) and they attack numerous tiles at once.
Corner-shooting is an extremely important tactic that is useful for all non-melee characters. Hiding behind a corner will hide you from enemies, but you'll still be able to see them. If you then shoot them with a shotgun repeatedly, they'll attempt to run at you (because they can't see you) while you constantly knock them back. Extremely powerful enemies can be taken out with a regular shotgun by relying on this tactic.
Lastly, shotguns will generally knock enemies back a square or two even at mid-long range. This is more than just important for corner shooting, you can use it to push tough enemies in to lava or barrel explosions too.
Angel of Shotgunnery/Shotgun-based characters
Shotguns are great weapons that almost every character will use at least once, so naturally a character who primarily (or exclusively) uses shotguns won't feel too restricted by their choice.
The main weaknesses of shotguns are not putting out quite as much 1v1 damage as rapid-fire weapons do and their poor damage type - shrapnel. Shrapnel damage doubles armor before dealing damage, so high-armor enemies will take little damage from shotguns, especially at long range.
This means that the correct master trait choice for Angel of Shotgunnery is almost always Army of the Dead - which completely solves one of your problems by changing the shotgun damage type to one that completely pierces armor. If you're just doing a regular game (or a challenge that does not restrict you to one weapon type) then you can go for a different master trait and rely on something else, such as a rocket launcher or BFG, for singular powerful enemies.
As a shotgun speciailist, you will almost certainly get Shottyman (a trait that is required for all of the following master traits), which makes weapons with long reload times suddenly not as bad. In particular the elephant gun - a regular shotgun with two power mods attached - works very well as long as you remember to never actually reload it manually.
Shotgun Marine - Army of the Dead
On Angel of Shotgunnery, this trait will allow you to take out more powerful enemies (such as barons and the cyberdemon) without having to worry about getting up close to overwhelm their armor. How handy!
Shotgun Scout - Shottyhead
Shottyhead (and also gunrunner, another good choice for shotgun-specialists) makes firing extremely safe. Your firing time with this trait will be about 0.3 seconds, which isn't quite instantaneous but might as well be if you're good at turn-timing horseplay.
With a tactical shotgun or assault shotgun your corner-shooting will be insane, 3 shots per second with no need to pump or reload beween each shot, you'll have to pass turns occaisionally just to give the enemy a chance to get back in range!
You can also bring around a double shotgun and fight up close in the open if you have confidence in your ability to manipulate turn order. You are extremely fast at performing any combat action, the only things that will be able to touch you are Arch-Viles and revenants! Because I can assume that you're probably not playing Angel of Shotgunnery due to the fact that you have this instead of Army of the Dead, they can be handled with the BFG or something, it's not like you need all those cells.
Shotgun Technician - Fireangel
Fireangel makes fighting in the open just a little bit less risky, and is also handy for people planning on lugging around a big rocket launcher. Your immunity to splash damage allows you to get out of situations that are otherwise impossible to dodge through - Arch-Viles and revenants will no longer deal miss damage to you and mancubi will only destroy items and walls with their haphazardly placed explosions, not you.
If you're planning on heading on in to the City of Skulls or Abyssal Plains for a missile launcher, its almost-perfect accuracy and your immunity to splash damage will combine to make a wepaon that's excellent for fighting in general. And just like your shotguns, it can be reloaded on the move with Shottyman! Make sure to pick up a lot of rockets to make full use of it if you do get one.