It's too simplistic to say, "either they're trustworthy or they aren't. If they are going to cheat, they'll cheat".
At the moment, this friendship seems to be threatening the OP's marriage. Either the DH doesn't know this and is sleepwalking into a scenario where he will cheat or, he does know, and is planning to cheat / already has.
If it's the former, the op needs to spell that out to him and make him realise how serious this is. He should not spend any more time with this woman.
It's possible that this isn't something the OW knows she's doing, (although her being a bit frosty with the op speaks volumes). But that doesn't matter; the op's DH has to put this new friendship to one side immediately and focus on what is happening in his marriage.
And that is either that he is deliberately trying to cheat, (so that needs to be sorted), or he isn't being careful and protective enough of what should be the most important relationship in his life. Either way, the OP clearly doesn't trust him altogether (rightly so), so that needs to be sorted too.
He needs to be made aware now, that this is a marital crisis. If he is made aware and ploughs on regardless, then it's obvious that he doesn't place as much value on his marriage as he does on a new friendship with some woman he has only met recently. And then the op then has her answer.
If she just shrugs, says, "if he cheats, he cheats. Nothing I can do about it", and the DH then sleepwalks into wrecking his marriage, it will be a long, painful, drawn out process where she watches her husband throw their marriage away and then hides it till HE'S ready to tell.
I would definitely try to take every possible, reasonable, step to save my marriage if it was under threat, but if my DH didn't do the same and let something like this continue, even after he'd been warned it was a deal breaker, I would obviously look into separating. And at least it would be quicker than the former.