Would it be okay to ask my ex not to text her new guy when we're together?
Is she still talking with him?
Yes I don't like the idea of her texting him when we're spending time together.
It is good to be concerned with an ex partner but it can come off as controlling if you are to forceful. Also, you should be open with your gf about spending time with your ex.
My girlfriend doesn't care either way.
It's good that your gf is so understanding and you have done what you can with your ex, but it is time to let them live their life how they see fit. Who's company they enjoy outside of your friendship is no longer in your hands, so leave them be and focus on your current relationship.