This is sounding worse by the day, so sorry OP.
He won't show you the messages because at the moment you are in that area of limbo where you don't actually know whats going on. Your gut says 'hey wtf', but without proof your head is questioning if your gut can be trusted. He wants to keep you in that limbo mode. By NOT showing you messages (which would clear all this up in a jiffy wouldn't it) you stay in limbo mode. Showing you them would mean he is showing you something he really knows you are not going to like and put you firmly out of limbo mode and into wtf mode.
Telling you not to contact her because 'it would cause issues at work' is his way of saying do not contact her because she has a totally different view of your relationship than the actual reality of your relationship. If he really didn't want to 'cause issues at work' he wouldn't be messaging at the bloody weekend and evenings, he would have proper work boundaries in place. There appear to be no boundaries in place with her at all. Sorry but that suggests she is blissfully unware he is happily married, or that she is the cause of so much anguish with you, the likely scenario is she has been fed crumbs of phrases such as you are inattentive, don't understand him, you've drifted apart, you are more like housemates now, and the good old classic of he can't leave you because you are depressed and on medication and he is thinking of the dc. Giving her the gold old, 'I'm such a poor, lonely, unloved victim, but i'm courageously doing the right thing' speil.
Him telling you that you are paranoid, controlling and lying to you about when hes with her outside of work is also paiting a lovely narrative for when he leaves, as he can recite how you drove him away with your unreasonableness, paranoia and in the end he couldn't take it anymore - even though he was the cause of it because he is being a gaslighting wanker.
It sounds like whatever you do now OP you cannot win. He has already created the groundwork in his head and he will start reciting the script as soon as an inkling of actual evidence is found. Which is will be found. And then he will rewite your entire history as the poor unhappy little man trapped in a marriage with a crazy wife.
I am truly sorry. He could clear all this up in an instant by being open with you, showing you the messages, even saying, yes call her because its important to me for you to be reassured so our marriage is safe. he is doing the opposite of all of those and blaming you for essentially butting into his little world of whatever he thinks is playing out or may potentially play out with this woman.
Just remember that. He has the power to clear this up instantly almost and stop you worrying. Why isn't he doing that? because he can't. he'd rather leave you in limbo mode until such time as he jumps ship or decides this other woman is not for him afterall. What sort of man purposely leaves his wife in that state of unknowingness. All the loving husbands i've come across would walk over hot coals to make sure their wife was happy and unworried.
I think you need to get your ducks in a row. Take the power back. The choice is no longer his.