hubby and coworker

My husband works with a married woman about ten years older than him. He is 24 she's 34. They've worked together in the same Dept for years. They are friends. And I'm ok with this. But they text each other, talk about work crap that needs done, but also about her kid and our kids and just talk in general. I began having a problem though when I noticed there were some days they would text all day. I asked to look at his phone the other day and he wouldn't let me. Because in the past I had and saw that he had called her sweetheart etc. and when she was having a bad day and needed extra help at work he had made comments such as "I'd do anything for you". I do not believe they would ever do anything physical and aside from her being a tiny bit attention seeking she hasn't really crossed lines aside from talking to him a lot and being too personal talking about her husband etc. My problem is my husband. And him offering too much emotional support or affection verbally I should say. I brought this up with her on fbk because I'm not comfortable with it and talking to him about it has gotten nowhere since he's still doing it and then as I said wouldn't let me see their conversation last time because he knew i would find comments like that that I wouldn't approve of. In my opinion it's borderline emotionally cheating. And according to her she's very happily married, that they are just friends, that yes they are very friendly but that I'm insecure and that he is loyal that she is just his friend. Of course she wouldn't see the issue because she's on the receiving end of his support in their convos. She told me I needed to talk to him if there's an issue. Which I try. At the end of the day he always ends up being too friendly and I feel like it's inappropriate, even disrespectful towards me. What should I do?