What is the actual therapy he is receiving, PartnerProblem? There is a big difference between CBT, DBT, ACT, CAT and others. If he's seeing a psychologist, that suggests he has a lot more going on than anxiety and depression - and his behaviour suggests that also.
If you don't know already, I think it would be helpful to find out what therapy he's receiving, and what illness/disorder the psychologist thinks he has. You can then educate yourself a bit more about what's really going on. I'm not asking you to share the details here, but just make sure you know what's what.
Then you'll be better placed to help him. And that help might be to allow the two of you some space so that he doesn't have distractions to get better. If he can get better. He may have a condition or disorder that will be around forever, and requires management over time. And if you know this, that gives you even more information to make an informed decision.
At the moment it comes across as if you have no real idea what's going on, or what's wrong other than "depression and anxiety". Truly, I think there's a lot more to it than that.
If I was you, I'd suggest going along to a session so that the psychologist can talk to you both.Not to divulge private things between them and your husband, but to help clarify matters, so you can more helpfully decide what to do.