Almost always an attorney fee award from the court is for reimbursement to the party for attorney's fees paid, not an order to pay directly the other party's attorney. You have a contractual obligation to pay your attorney. As a general rule a fee award is enforceable through the contempt powers of the court, but be award that impossibility is a valid defense to contempt. The fee award is also collectible as a judgment, and your attorney may be willing to assist in doing so but your attorney may be entitled to earn an additional fee for collection services.
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You owe your attorney the fees. The ex owes the fees to you. You will have to collect them from her. If the judgment includes fees, you can execute on it, or your attorney can take those steps.
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Unless your attorney agreed to wait to receive payment from you, you agreed to pay the attorney and that is your obligation, so yes you owe the money absent some specific agreement to the contrary. However, if you received a judgment in the lawsuit, you should be able to execute on the judgment and collect money from your ex whether he wants to pay or not. You should discuss with your attorney (or perhaps a new one) what your rights are in this regard.
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