If you were served with a restraining order then I may say to not close the account. You need to take your pleadings to a attorney to ask this type of question. Sounds like a child support modification but giving you a green light to close accounts would not be wise until your file is reviewed.
In setting child support the court has the right to look at all resources you have. I think closing a bank account while the process is going on looks like you are hiding assets, which could come back to haunt you. If you have a straight paycheck job, the AG generally calculates child support based on your earnings. If you are self employed it gets a little trickier and that's where your savings comes into play. You did not say if this is an establishment of parentage or a modification of a prior order. Those issues will change how you approach this. There are lots of good Harris County attorneys who do family law here on Avvo who are reasonable. Look for someone who spends a lot of time in AG or DRO court and get a consultation to figure out your best strategy. Good luck.
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Sure - you can - but why? You'll have to disclose your cash as well as your bank accounts. Consult a local attorney - but it will look like you're trying to perpetrate a fraud if you're closing accounts and moving money around for no legitimate purpose.
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Your checking accounts normally don't affect the amount you pay in support unless you owe arrears (back support) then they may attach your accounts to pay back child support.