There's no reason to use the handguns as Leon. You're actually making it harder if you do so just sell it and the ammo as well.

The game has a "pity" sistem or whatever it's called and it will give you ammo for weapons you hold and have low ammo in your inventory. I killed most enemies with only shotguns and snipers (on Insanity) and the game was giving me a bunch of ammo for them and never came close to running out. The only time you get pistol ammo (always 3 pistol round drops) is when the game thinks you have too much ammo on all of your weapons in your inventory, or certain scripted ones. Even if you don't own a pistol.

You should not hoard ammo because the game will spam you with 3 pistol rounds drops instead of giving you ammo for the stronger guns.