Nikki, I think you need to tackle this woman and your DH.
I had a similar situation a few yrs ago.A divorced older woman and outgoing friendly husband working together. I was uncomfortable with her manner with him, the amount she would text/phone etc, he was oblivious though and initially felt I was making a fuss out of nothing.
Alarm bells rang when I read in your post that she had said that others were saying her and your DH were having an affair as this is what happened in my DHs case.
Luckily he realized it was an opener for him to say something along the lines of how nice that would be. He didnt and instead came home and told me I was right and she was after him.
The following day, she said how much she liked him and tried to kiss him!! He told her as nicely as he could(too nicely knowing him!!) that it was not what he wanted. I think this may have been the point at which it became an obsession for her. She then bombarded him with emails telling him how she couldnt live without him etc etc she twisted all the things he said to repel her gently by impling it was cos he was under the thumb.
Eventually he had to be utterly blunt and rude and cut off all contact with her (which was difficult as they were still working in the same place) change our numbers and email.
Basically I think that she may have been feeling vulnerable(she sounds just like the woman you describe tbh, newly separated too) and took how he is with everyone,(friendly, makes people feel good about themselves)personally, and then couldnt accept that he didnt want her when she made it clear she wanted him.
I'm not saying this will def happen, but just reminded me so much of our situation that I had to post. I was lucky that DH saw what she was doing cos he is a bit clueless about that kind of thing. He only saw it I think because I put the possibility in his head otherwise he might have ended up in a far worse situation as she took not saying anything to be a positive response iyswim.
I think you need to ask your DH if he is getting so much out of this friendship that he doesnt mind upsetting you to continue it. She is attempting to oust you I think, and you have no idea what mind games she is playing when you aren't there. DH remembered (after she had shown her true colors)other things she said designed to make him think poorly of me(I don't suppose you were allowed to watch the footy, cant you escape for a drinky etc).
Having said all that maybe she is just a friendly woman. Even so you need her to understand she is overstepping your boundary.