K.S.
First of all a HUGE hug to you. I can understand why you are so stressed out. You most definitely need a break and it is unfortunate that your husband is not mature enough to understand someone else's point of view. I certainly advocate that married couples work together to solve problems but it doesn't sound like he is open to discussing options. I would try one more time to sit down with him. Present an open ended question something like, "I am at my breaking point right now and I need your help to figure out what to do. I can't continue this way. Are you willing to listen and then help come up with a solution or do I need to figure this out and solve it on my own. Continuing like we are is NOT an option!"
If he says he will help then sit together and make a list of what is working well in your lives and a list of problems to be solved. Start with what is good and use that as a basis for moving forward. Then pick one of the problem areas each week to discuss together. Set a regular day and time that you use to discuss family matters. And make sure he gets an opportunity to include things that he would like to change.
If he is unwilling to help, you have a problem. Continuing as things are is not a healthy option for you or your kids or your marriage. Even if he won't seek counseling, you need to for yourself so that you have that support.
From a problem solving perspective...are there other moms around you that would be interested in setting up a babysitting exchange so that you could at least get some breaks? Have you checked into any job options and what your pay would be as well as what daycare options and that cost? If you write out a summary of the finances you may be able to prove that it makes economic sense for the family. I don't know your husband's military duties right now but if you have times when he can care for the baby and you could do a home based selling business like Tastefully Simple or Avon or other, that would give you a chance to get some adult interaction. That would also give him perspective on how hard it can be with a screaming child.
Good luck and keep us posted.