I am in my 2nd yr of doing taxes and know only enough to know of the claimings. If your court papers say that he can claim them for the upcoming tax season, he does have the right to claim them. If he files before you and claim the children using efile and you file after him claiming the children also there will be a delay in your taxes. One of you will need to provide the IRS with documentation of the children (living, school, and various other expenses, but he will have the papers showing that this would be the year to claim the children.
This inturn will reject you from using the efile until you requalify (can take upto 3yrs) with the IRS. You can see if he will sign a waiver for you to claim the children for this year, but this will have to be mailed in and efile cant be used (ask your tax preparer before you file, he should give you a form that your ex will need to fill out) and make sure that he (your ex) agrees with this also.
Your best bet (and I am sorry, this may cost you money) is to take him back to court and prove that he doesnt pay on a monthly basis, so he shouldnt have the option just to claim them. On a yearly basis you will have the right to decide who gets to claim.
Hope this helps.