It sounds as though you have done, mostly, the appropriate things in general in regards to the illegal/fraudulent usage of the children's SSNs. I would suggest writing letters to the relevant utility companies confirming any telephone calls and with proof regarding your children's SSNs. Send these certified, receipt requested, and keep the receipts with copies of those letters.
Have no criminal charges been filed against him? I do not practice in Louisiana, but this sort of activity could be felonies where I am. (The victims could be either the ultilities or your children, depending on the particular charge.) Unfortunately, you cannot force the police to file such charges, even if you wanted to push it that far. The FTC site discussed this issue.
Make sure your family court attorney has copies of all documents related to the use of the children's SSNs and is completely up to date on this issue. Your attorney will let the court know this has been happening, and that the electric has been shut off at your ex's.
Whether you get custody as a result of your ex's actions would depend on whether the judge decides that your situation is better for the children. So you also want to bring whatever proof the court might need which would address whatever concerns the court may have concerning you as possible custodial parent. (I have to assume there is a reason why the father is the custodial parent.) Discuss this with your lawyer in detail and in private so you can be fully prepared. Discuss also whether it would be better to have your kids go to foster care or for you to pay the utilities (possibly as part of your child support), if needed, if you are unlikely to get custody right now.
If you do not have a lawyer for family court -- get one. You need the help in order to help your children.
DISCLAIMER: I do not practice in LA. This answer to a short question is provided solely for general informational purposes and based on common sense, general legal principles and court practice. This answer does NOT constitute legal advice, create an attorney-client relationship, or constitute attorney advertising.