Some ways to increase damage overall, which will also increase your damage with shotguns:
(Note that for every kill, the 30 second timer restarts)
There are also other legendary modifiers, but the list is quite lengthy and a lot of them are only applicable against certain creatures. Take a look at the Wiki for some more.
You already stated you know about the shotgun perk cards, but you may have thought to yourself, why are there three different shotgun perk cards that all do the same thing (Shotgunner, Expert Shotgunner, and Master Shotgunner)? I think the main reason is that you can get a higher damage percentage for less SPECIAL points if you use three Rank 1 cards instead of one Rank 3 card and one Rank 1 card. For example:
- Master Shotgunner Rank 3 + Shotgunner Rank 1 = 30% more damage for 4 Strength points
- Rank 1 Shotgunner, Expert Shotgunner, and Master Shotgunner = 30% more damage for 3 Strength points.
You are only allowed a total of 50 SPECIAL points, and up to a maximum of 15 in each attribute. So by allocating your perk cards in this manner, you can cut down on the amount of SPECIAL points you need, thus allowing you to allocate points to different perks. The same thing goes for the other perks cards like this, such as rifleman, and the melee weapon ones. Of course to get maximum damage output, you would need Rank 3 in all three of these cards, but that would require a hefty 9 Strength points, leaving you with at most 6 more Strength points to allocate to something else, and that's only if you have 15 Strength.