The first perk cards you 'find' are the ones in the vault when you start a new character. Those are the only ones you can find in the world and the rest are unlocked when you level up. This is either by simply leveling up and unlocking said card or by getting them from a card pack.
Each time you level up, you'll be able to pick one point to increase from your SPECIAL, and then choose a Perk Card. Each SPECIAL stat has their own associated cards, and each time you level up, new cards will be added to pick from (but not all categories get a new card every level). You also do not need to necessarily pick a card from the stat you increased. Any cards not picked will remain locked until the next time you level up, at which point you can choose another card.
S.P.E.C.I.A.L
Each SPECIAL stat can have up to 15 points. SPECIAL points are earned with each level up until level 50, and your character starts with one point in each SPECIAL. It's important to note that even though you stop gaining SPECIAL points at level 50, you can keep getting more cards every level, and there doesn't seem to be a level cap at all.
Perk card packs
Perk Card Packs can only be obtained by leveling up, and are not for purchase. You'll earn card packs at every other level starting at level 4 until you reach level 10. After that, card packs are given every 5 levels.
Sharing perks
Once you put at least three points in Charisma, you'll have the opportunity to share active perks with the rest of your team. As long as your Charisma is three times the point cost of a perk card (for example, Rank 1 of Green Thumb costs 1 point, so you could share it with 3 Charisma. Rank 2 costs 2 points, so you would need 6 Charisma, and so on) you'll be able to let your team reap the benefits of your card, and use it to plan effective strategies to synergize available perks.
Source
As of now - on Fallout 76 Patch 11 - the current meta is to use weapons with the bloodied legendary effect. I am particularly partial to a bloodied Handmade Rifle, as are many other players.
Refer to this post and this special build for a general guide of what to use. Lone Wanderer is critical if playing alone, any other perks to boost gun damage are important too.
As a general guide though, the gun alone isn't what gets you to the optimal build, the legendary effect on it is. Anything bloodied is going to be good. Two-shot and explosive has been recently nerfed, but a two-shot explosive handmade is now much cheaper due to the nerf and still very effective.
As I said in the comments, you can farm springs from clipboards in the National Isolated Radio Array. Clipboards are abundant there, I'd recommend setting up camp nearby.
Best Answer
Some ways to increase damage overall, which will also increase your damage with shotguns:
Bloody Mess Perk (Luck):
Adrenaline Perk (Agility):
(Note that for every kill, the 30 second timer restarts)
Small Guns Bobblehead:
Sneak Attacks:
Chems:
Instigating Legendary Weapons:
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:
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.