A549

There is certainly room for confusion in the way the Full Attack section is written in the MM and others. It arises, as far as I can tell, over a misunderstanding of what constitutes a creature's primary weapon. While 'weapon' is written as singular it does actually encompass multiple natural weapons of the same type (in this case claw).

If you only have 2 claws, then claw is your primary weapon, and you add your basic (1x) Str bonus to both attack rolls and damage rolls with (both) those claws. Note, even if you only use one of those claws to attack (for example making an AoO) you will still only get to apply your basic (1x) Str bonus to any rolls.

If, for some reason, you chose to wield a weapon, this becomes your primary weapon, and your remaining claw becomes, assuming the weapon you're wielding is 1-handed, a secondary weapon. At this point you would add your basic (1x) Str bonus to your damage roll with the weapon, but only half your basic (0.5x) Str bonus to damage done with the claw (as well as suffering additional penalty of -5 to hit with the claw - barring any mitigation from feats).

There are numerous examples (of monsters) in the MM that back this up.