If a card just says "pay 1" then you can pay one of anything. If for instance it had the Agency logo in the cost, then you'd have to pay with that type of domain. Normally though, this is not the case.
It's a nice weapon, sure, but there are drawbacks.
First, leaving domains open to pay for shotgun is not free. Every open domain is a character, support, or event card that maybe you didn't play. OK, sometimes one of your three domains is 1-cost and might not pay for anything that great, but go beyond that and you're giving up real card playing power. If you're often finding that you cannot play more than one card, I would double-check that you're doing your resourcing right. (See below)
Second, if you're committing several characters to "protect" him - as you said one to go insane and one to die to combat, that's a sizable amount of commitment to that story. Perhaps your opponent will just let you have it rather than walking into twin shotgun blasts, and then with a good chunk of your characters exhausted walk all over you in their own story phase. There are plenty of times where I'd trade my opponent some success tokens on his turn for an equal or (likely) greater number of success tokens for myself on my own turn. Don't expect them to commit their best guys to the story you pick just to get plugged, or to show up without a counter-plan
Third, any attachment makes a character a big target for wound/destroy/sacrifice/insanity/uncommit effects because you're getting a 2-for-1. Heck, if you time it right maybe more than that, you could be netting a character, his shotgun, AND the use of those domains your opponent left free that he could have spent on something else. Even just exhausting him to keep him out of the battle is big because you're also nullifying the domains they didn't spend that maybe could have been doing something else useful.
Wounds can also be countered by Toughness, or in some cases Invulnerability. Sometimes all it takes is one fight where you don't kill the thing you expected to and your guys get killed.
Bottom line is that the Shotgun is useful, but I wouldn't consider it overpowered. It adds no stats, you see it coming because it's a support and not an event, and you have to pay every time you want to use it. Deal with the character, and you're also dealing with the weapon, and the shotgun doesn't help protect against anything for instance like the Springfield does by raising your Skill (thereby making you more resistant to some cards).
OK, back to the "can only play one card" thing... There are a couple of potential causes you might see this.
1. You might be resourcing incorrectly in some way. You start the game with 1-1-1 and every turn you can add one resource. So, for instance you may grow 2-1-1, 3-1-1 / 2-2-1, and so on. Typically you always want to resource early on, and at some point you may stop if you have enough resources but that's generally not for at least 3-4 turns and maybe more.
2 You might be including too many expensive cards in your deck? You should probably have a good amount of cost 2 stuff in there, and a lesser amount of cost 1 & 3, not so much that costs 4+, just as a rough guide. If you put in too many expensive cards then it makes sense you'd be unable to afford multiple cards but overall it makes your deck too slow.
I'd also like to add that if you're often finding yourself only playing one card per turn, how can you afford to send 2 guys in to protect Mr. Shotgun? It seems like this might be committing most of your force, leaving very little for defense?