Oh OP, I’m so sorry you’re going through this and particularly whilst you are pregnant. You have to do what is right for you and none of us can judge that. Unfortunately, lots of us have been in your position, though, and can try and give you advice from the perspective of being further down the road.
The lies that go with alcoholism are really really corrosive to a marriage and I would say were a very big factor in ending mine. I heartily recommend getting yourself some counselling for this, particularly with a counsellor who understands alcoholism.
My best advice is to try and get some super charged boundaries around what “dealing with the problem” looks like to you. You won’t want to hear this but I can’t tell you how many times I heard the this is it, I’m getting sorted speech. Usually it was accompanied by a huge effort of will power to genuinely try. Unfortunately, will power has no effect with alcoholism and those efforts are doomed to failure. It is very normal for an alcoholic to tell you everything you want to hear but they are driven by compulsion that talks a lot louder than you can.
You say your h has accessed some support services. Has he been really open with you about what this is? Is he attending aa meetings (people are often recommended to try and attend almost daily in the early stages)? Getting an a sponsor? Etc etc. An alcoholic who is truly desperate to change will throw themselves into this. A half hearted effort won’t do anything- it’s a sure sign he’s going through the motions to get everyone off his back.
Even with lots of support, you should be aware that the statistics for people successfully getting long term sober are really depressing. I’m not saying that to put you off giving him this chance but to be eyes wide open that you may find yourself back here repeatedly (sorry). You need to think about where your “enough is enough“ point is because it is easy for this to become a (warped) version of normal without you really realising it.
Re your children. You cannot trust him to do childcare or drive them until he has proven his continued abstinence for a good long time. There will be a very high chance of relapse, particularly during the first year.
Gosh, this probably sounds really preachy amd I’m sorry if it does. Above all, make sure you keep your boundaries really strong and do get specific help for yourself (counselling/Alanon) as well as from your family and friends (they sound great).
I ended up back where you are now so many times before I ended it. I should have done it sooner and I think you’ll find a lot of others with the same experience would say that too.
Really good luck and look after yourself.