My ex is badmouthing me to the children and I’m worried they’ll start to listen to what she’s saying. How do I handle this?
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Continue to reassure your children that you love them and that although you no longer live with them, you will continue to see them regularly and that you are only a phone call away.
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Whatever you do don't a) bad mouth her back through the children or b) discuss your feelings about what she is saying with your kids. Keep them out of your issues with your ex. All you can do is communicate to her that it isn't best for the kids. (And make sure you don't bad mouth her!)
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I agree with the comments above and below, your steady reassuring love is the best response as far as the children are concerned. It might help you if you find a listening ear you can trust to talk out any anger or frustration you feel about your ex's badmouthing of you. This is likely to prevent it spilling out when you see her or have the children.