tl;dr I am completely emotionally and mentally exhausted, as well as significantly physically exhausted, and have completely burned out this last week with respect to my young children and my graduate research work. I need a vacation with my spouse without my kids. Problem is, we've never left our kids (3.5 years old and 10 months old) for an extended period and don't know how they'll behave, and the 10 month old isn't weaned yet. I don't know if I can wait the two months before the baby gets weaned (as is the current plan) before having a vacation, and the baby has never drunk from a bottle so I don't know if we can switch her over to formula.
How can I take a vacation from my kids when it seems not possible to actually leave one of them behind?
This last week, I crashed and burned out when my 10-month old baby (who had just started sleeping through the night) regressed and decided to start waking up early again. I developed a depressed mood and, although I feel physically and emotionally exhausted, am unable to fall asleep earlier to compensate for waking up earlier. I am in the middle of an aggressive research-oriented graduate program which has come to completely overwhelm me as I crashed. I have lost all motivation for my research, and almost all motivation in my home life. I had no idea I was so close to crashing until it happened.
I have a 3.5 year old as well. He is loud, my baby is loud, and we live in a small apartment so it is impossible to escape the noise anywhere. Even on the days I feel like I can sleep in and try to recover (my spouse taking sole care of the kids during that time), the inescapable noise makes it impossible. I have grown very short and impatient with my older kid as part of my burnout.
I don't have anywhere to escape. My work is stressful and I have no motivation for it, so my office is not relaxing. My home has no sanctuaries in it because it is so small. I don't really have any hobbies any more because of the time my research and family takes, and I don't feel like I have any friends I am close enough to to just hang out with. Also, I hate the US state I'm living in right now, there is very little available of what I am used to doing (hiking, outdoors etc.) and many other reasons.
edit: answering some more questions. I probably wouldn't be able to hire day help, but that wouldn't help anyway since I need someone to take care of the kids in the evening and overnight, and what I really need is to be in a different place than the kids, not just have someone else take care of them. Baby is eating foods now, probably about half-and-half food and breast milk. I have tried earplugs, they don't work very well on the high-pitched noises that are the most troublesome. I run fans too while I'm sleeping, still not enough. My apartment is just too small (650 sq. ft.). I think I can technically take a break from grad school, but my clock will still be ticking towards my deadline for graduating (summer next year) and I'm already so far behind on my research due to the fact that I have kids that that by itself is a major stressor, let alone considering taking major time off. I can enter a special semi-student status if I need to keep working on my dissertation past what should have been my graduation date, but that presents a new set of problems (obtaining funding, for example) and extends the amount of time I'm in this life situation. We live far away from family so it's not an option to leave kids with their grandparents.