Hello sorry you are having a bad time,
It is common for people to find it difficult when partners confide in others, in fact me being on here is something that my wife struggles with. But it can be very healthy, even just from the point of view of getting things written down. But flirty behaviour I can understand is more unnerving.
However therapy can come in many forms, if he is functioning well in the company of others it could be a good thing, or at least one less thing to worry about. I spect he is puttting up a front with you, holding in emotions, so will be tense around you, all you can do it try to let him know that he can say anything he wants to you, and not to be ashamed of how he feels.