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You can most certainly grow claws while Wildshaped, here's the rules text to back it up (found in the Magic section of the Core Rulebook under Polymorph, or found here in the PRD)
While under the effects of a polymorph spell, you lose all extraordinary and supernatural abilities that depend on your original form (such as keen senses, scent, and darkvision), as well as any natural attacks and movement types possessed by your original form. You also lose any class features that depend upon form, but those that allow you to add features (such as sorcerers that can grow claws) still function. While most of these should be obvious, the GM is the final arbiter of what abilities depend on form and are lost when a new form is assumed. Your new form might restore a number of these abilities if they are possessed by the new form.
There might be some controversy regarding whether the slam and claw attacks are trying to use the same limb, and whether that prevents them from being used together in the same round. That's a harder one to rule on though. The relevant rules text, while it seems to imply one limb, one attack, doesn't nail it down precisely either:
Natural Attacks: Attacks made with natural weapons, such as claws and bites, are melee attacks that can be made against any creature within your reach (usually 5 feet). These attacks are made using your full attack bonus and deal an amount of damage that depends on their type (plus your Strength modifier, as normal). You do not receive additional natural attacks for a high base attack bonus. Instead, you receive additional attack rolls for multiple limb and body parts capable of making the attack (as noted by the race or ability that grants the attacks). If you possess only one natural attack (such as a bite—two claw attacks do not qualify), you add 1–1/2 times your Strength bonus on damage rolls made with that attack.
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You can make attacks with natural weapons in combination with attacks made with a melee weapon and unarmed strikes, so long as a different limb is used for each attack. For example, you cannot make a claw attack and also use that hand to make attacks with a longsword. When you make additional attacks in this way, all of your natural attacks are treated as secondary natural attacks, using your base attack bonus minus 5 and adding only 1/2 of your Strength modifier on damage rolls. Feats such as Two-Weapon Fighting and Multiattack can reduce these penalties.
It's pretty clear on manufactured weapons / natural weapons using the same limb, but not so clear on how two natural weapons ostensibly delivered via the same limb would work.
To me, that seems to fall in the iffy, ask your GM, area. You could argue that you slam with the elbow / forearm and claw with the hand.
Apologies for the thread necromancy, but I was trying to answer the same question myself and figured it was worth sharing the result.