Wendy I can feel your pain just by reading this. You are in an awful situation and I'm glad your friend is being supportive and also your family.
You know only too well what a lot of us think on here and I wondered whether you had considered that this might actually be a good thing (if there is an OW)? I know that sounds strange, but if you really do want to fight for your marriage, it might help you to think that up until now, it has been impossible to compete. Most affairs are about escapism, they very often happen at key life catalysts (illness in family, milestone birthday looming) and for a time, people in the grip of them go through a kind of temporary insanity.
For a time, while these things take place in an illicit bubble, it is impossible for the unknowing spouse to compete. They have no idea there is anything to compete for and so they go about their daily lives as normal, knowing deep down that something is not right, but not being able to put their finger on quite what it is. It is not a level playing field and therefore the comparisons many adulterers make between the affair partner and their life partner are odious and unrealistic.
Normally, once the shit has really hit the fan and the affair starts to intrude on reality, people "wake up" and come to their senses. Sometimes this takes a while though.
If he really has decided (with no outside influences) that he wants out of the marriage, then I think you should cut your losses and accept that.
You might need to think "Ok, if it IS an affair, what would I want if he ended it and asked to come back?" Really consider that question. If the answer is that (with conditions) you would have him back, then there might be some hope for this marriage. But as long as he maintains his story of there being no-one else, you can't deal with the reality.
My advice to you, either if he wants out for good, or just THINKS he wants out because he's having an affair, is the same, actually.
Continue what you're doing in making him realise the seriousness of this. In a few days time, book an appointment with a solicitor and tell him (by text) that this is what you are doing. Start making noises about finances and future plans generally. Act tough, even if you feel anything but inside.
Once he sees that you are taking control of your life and have appeared to have accepted that it is really over (you might even want to feign relief that you can now move on with your life) it might just be the thing that causes him to "wake up".
Take support wherever you can over the coming weeks and stick to your guns Wendy. View this as a task that you must accomplish - you see, by taking control at last, you are now in fight mode and at the moment, you think he is the prize. You might just find that at the end of all this, you don't want him, though and that will be his (and so many other men's who have gone this way) tough luck.
Thinking of you.