Is it even worth picking up weapons in the campaign?
Solution 1:
I don't pick up a second weapon beacuse its better then my first. I grab weapons so I can build an arsenal to deal with any situation.
For instance, if I'm doing a sniper mission the first thing I will do is replace my sidearm with the first shotgun, smg, or assult rifle I found. Carrying multiple guns gives you the versatility to always have the appropriate gun for the appropriate situation.
Finally, don't forget that its faster to switch weapons then it is to reload. Carrying a second gun you enjoy can make your charge 2x more awesome!
Hope this helps
Solution 2:
I swap my secondary with enemies' weapons. Note that, they have very little ammo left in them, most probably around 100 rounds, whereas the gun you started with will have something like 500 rounds. It would be a bad idea to replace the primary.
And, if the secondary is a sniper (e.g. the mission in Africa, where Price, Soap and Yuri go to do stuff), you might need it later in the mission. Swapping it wouldn't help. Although in MW3, I found another sniper when I was required to do some sniping. This wasn't the case in MW2.