It's hard when the people we love the most take out their anger or fears on us. Unfortunately, it sounds like he's had some pretty terrible behavior modeled for him as he grew up. But you are right, the past does not have to dictate our future and it doesn't justify how he is treating you.
It sounds like he is mentally, emotionally and verbally abusing you and sometimes those wounds can run deeper than if he'd actually get physically abusive. Accusing you of cheating is probably another way for him to push his insecurity onto you. The problems he is having really do have to deal with him and HE will have to be the one who wants to work hard to change himself and deal with his past demons.
If you love him and want to continue to work it out (and as long as he doesn't become physically abusive to you or your children) then I would just try to support him the best you can while he goes through therapy. The anti-depressant medication from your doctor should help him be a bit more balanced.
It sounds like you don't want to give up on the marriage and just divorce so I would fight for what you DO want. I would recommend setting some boundaries for yourself and pertaining to your children while you guys both are working through this. For example, HE needs to start telling the truth about your finances, fights are NOT to happen in front of the children and you both have the right to walk away from an arguement for a "grown-up time-out" until you can come back together and talk in a more civil, communicative way. I would be concerned that eventually if he doesn't deal with the issues than he could start taking things out on the kids too. It really is your first responsible to take care of yourself and your children. He is a grown man who you can love and support but unfortunately you won't be able to change him or "fix" him.
You are also right that the children can sense that something doesn't feel "right" or "safe" right now and that does effect who they are. But love does cover a multitude of sins so since I personally believe in the covenant of marriage than if I were in your situation right now...I would try to make it work until I had exhausted every avenue. It could be that your marriage becomes stronger after going through the tough times together. However, if you do feel like it's not worth it than give yourself a couple months, if you can, before making your final decision. I wouldn't rush into any decision at this point and regret it later.
Just remember, you are worthy of love and respect and it's the right thing to expect that and nothing less from your spouse! Unfortunately, it sounds like he may not be able to give you what you need and deserve if he doesn't have it to give. And, start praying! I would be covering your marriage and your children in prayer. The power of prayer is amazing! And, don't be afraid to reach out for more help. It is time for him to not just throw fits and accusations and actually walk the walk to go to therapy and get the help he needs to have a more joyful, peaceful, fulfilling life. Sometimes people just don't seem to have a clue how much their actions affect the lives of those around them. And, if he changes his mind and never goes to therapy than I'd be thinking about some ultimatums because this isn't going to just get all better without help. It's not something to sweep under the rug. The fact is, healing past wounds and learning how to deal with your feelings is an ongoing process for all people and it won't just get better overnight. Blessings to you and your family!