I too have never been an FPS player. I am just not interested usually in twitch gaming. I've long been of the opinion that I would svck at FPS. An RPG such as Fallout however is a different story. I find VATS breaks my immersion and I never use it, ever, for anything, including finding enemies. I play as if VATS did not exist. I do play stealthily however, and I am not sure how well I would fare if I did not. Oddly I find I am 'not' svcking doing this in Fallout.
I would recommend you concentrate on your rifle and a combat shotgun (ditch that double right fast), this mainly so you don't have to spend twice as many perk points keeping your damage maxed. For the rifle, trick that out for long range aimed fire. For the shotgun, give it a short barrel and no scope, just a reflex sight (also anything to reduce recoil). You want a short barrel because you 'want' the wider spread. You don't really need more than this. You might want to keep a melee weapon for riffraff you don't want to waste ammo on, and a fatman for behemoths and the like, but a rifle and shotgun should do the job in 99% of the situations. The rifle obviously is for when you can engage at a distance. The shotgun is for when things are close and moving fast.
Other things you want are a very good framerate and a good mouse. If you are not getting a solid 60 FPS in the thick of battle, you should be lowering your graphics settings until you do. Some FPS gamers would insist on 90 FPS. As I said before, I am not any sort of seasoned FPS player, but by having a good FPS and responsive mouse, I am able to resolve most fights with just my rifle, even while the enemy is charging me.