My take is that when you stop the ILYs, he goes back to pursuing you again. He needs to hear it for reassurance that you're there for him. But he's cakewalking by still being in contact with the OW. It may very well just end soon if he's trying to end it. Or not. You need to keep your eyes and ears open.
The reason they say not to say ILY, is because by doing so you've made the WAS think again of why they don't love you, and that's not the thought you want a WAS to be reflecting on. It's also pursuing behavior. That doesn't sound like it's material to your situation at this point. He's the one initiating the ILYs.
About the phone calls... has your H told you he has stopped contact with the OP some time ago? If he hasn't said so, why not ask? If he asks why you ask, you needn't tell him that you've been snooping. You could tell him that you do not want to have maintained a wall of secrecy and wish to know details as you feel the need to.
