Hi NoseyHelen, I couldn't read your post and not reply. I am a counsellor and do not presume to know about your relationship. I would like to lend some support (apologies for the long post).
Firstly, I'm sorry that you have found this out about your DH. It must be a shock, especially if it's come out of the blue.
I will go against the grain and say yes, I think that your DH is not being truthful but does not seem to have taken the final step of actually going to the Escort Agencies. My gut instinct says he has thought about it but not gone through with it. The multiple searches suggest that he has not made up his mind and might be using them as a bit of escapism(for now). If he had gone to an Agency, he would probably initially return to the same one, because he would feel that he had gotten away with it and had adequately covered his tracks.
You both need to talk, is there someone who could take the kids for a few hours to give you time to talk/shout without the fear of the chilidren overhearing? I agree with you about finding a relationship counsellor and about your concerns about what happens if he refuses to go. How you pitch the counselling idea is very important because if he feels that he is going to be under attack or blamed, the chances are he will refuse to go, which leaves you in a tricky situation. It seems unfair that you have to be the one to pussyfoot around him when he is in the wrong but it is important to keep dialogue open, especially because you are worried about your children. As upset and angry as you are now, it might be an idea to try and organise your thoughts before talking to him which may help you focus. He knows that you know and is probably bricking it and thinking of how to get himself out of this mess. He knows that he has been caught out and has made matters worse by lying about it being a one-off and then deleting the history. The next logical step would be for him to be on the defensive. Ultimatums and threats could back him into a corner, achieving the opposite of what you want.
I would suggest that you write yourself a list of questions, if you can, to help you to clear your head, on the lines of:
a)What do I want to do? Do you want to stay with him? If yes, then what needs to change?
b)What you think has gone wrong in the relationship (you mentioned in OP that you are not intimate-have you spoken about it? Do you miss intimacy with him? If yes, chances are he misses it too. What would you would like to do to get it back on track? It might be the last thing you want right now but is something that will need to be addressed if you decide to stay with him.
c)Is there something else going on with him? Stress, worry, self esteem problems, weight issues etc? This does not excuse his behaviour in any way but might help get to the root of all this.
d)Has he been a good DH/ father before this happened? If he came clean, would you be able to forgive him and allow him to earn your trust again?
I agree with what someone else said about going to a GUM clinic, even if it's for piece of mind. Whatever the outcome of all this, this could be the start of a new chapter in your relationship.
I wish you all the best during this difficult time. I'll follow an earlier example, if I may and also give you a hug x