If I enroll in my company's dependent care FSA for the full $5000, but my full year child care expenses are $12000, am I still able to claim the $3000 tax credit ?

If you have one child the FSA money you already used is pre-tax.  You do not get to double dip and get another $3000 childcare credit on top of that.  If you have two or more children in daycare, then you could get up to $1000 of the childcare credit -- the difference between the $5000 of FSA money and the $6000 childcare credit limit for 2 or more children.