Do we stay where we are and deal with the frustration or buy a house?

TL;DR Our living situation is becoming very frustrating but we don’t want to haphazardly jump into a stupid financial decision.
My husband (23M) and I (22F) feel super stuck and don’t know where to go from where we are now. I don’t want to sound ungrateful because I am so thankful to have a roof over our heads and a safe place to live especially in this economy. In February our lease ended at the townhome we were renting and we moved in with his mom and stepdad to save for a house (they offered this option). We went from paying $900 in rent to $400, which includes our share of the utilities. Financially, we’re still living like we were before and putting the difference ($500) aside into savings. We also received a much larger than expected tax return and after paying off our credit cards, we saved the remainder However, although we’re able to save way more than we were previously, we are pretty miserable. We have two nine month olds (twins), my husband, and myself, and we all share one large bedroom. We have access to the kitchen/our own bathroom (we have to share a shower), and shared laundry. The main part that sucks is that my husband’s stepdad, who sleeps very lightly, works third shift and refuses to use any sort of earplugs or sound machine. So as you can imagine, if I take our girls into the kitchen any time from 7am-3pm ish, he wakes up. The slightest sound wakes him up, so two screaming children definitely do. He always makes it clear that he’s annoyed when we wake him up, but it’s not like I’m doing it on purpose. My MIL says he can get over it and I should do whatever I want, which I get, but his grumpiness and passive-aggressiveness is starting to weigh on me and just getting really frustrating overall. 
In addition, I am generally the only person who does the dishes (and the majority of the day-to-day cleaning), along with my husband but he works long hours. I have never once seen my MIL’s husband do them and I know he doesn’t, and my MIL says I can either leave them and she’ll do them on Saturdays which only actually happens sometimes/have her cleaning lady do them, or if they bother me (which they do because I can’t cook when they’re in the way) I can do them. Btw, cleaning lady doesn’t come consistently, only when my MIL can afford it. They eat out or order doordash most days and so they don’t care about the dishes getting piled up. On the other hand I have to cook most days because we can’t afford to eat out and save what we do. Most days if I don’t do the dishes, I can’t even access the faucet.
They also have a large dog who sheds EVERYWHERE and is not trained. The most they do is sweep the hair into a pile and let the cleaning lady deal with it. He jumps on the counters when I’m cooking and takes food out of our girls’ hands. He does not listen when I tell him no. They’ve tried a buzzing/beeping collar but that does nothing. Dog also barks at every single vehicle that pulls in the driveway and it always wakes up our daughters if they’re asleep. My MIL and her husband say it was our decision to move in with them and we need to deal with him being that way, which is true, but still equally frustrating as we’re not bumming off of them and we’re paying the agreed upon rent. Additionally, their dog is crazy and freaks tf out when he sees our cats. We have to make sure our door is shut the right way or he’ll burst into our room and attack them, which happened once and if my husband hadn't been there to stop him I’m confident he would’ve killed our cat. They’ve had other pets before so we didn’t realize this would be such a huge issue. Just a lot of stuff that we would’ve completely avoided if we’d have known beforehand.
I stay at home with our girls, don’t sleep much, and was recently diagnosed with a condition that greatly affects my energy levels and makes it really difficult for me to engage in anything more strenuous than a walk, so besides this entire situation, I’m already stressed enough. I’m also breastfeeding two babies and that’s mentally and physically draining in and of itself. My husband is a huge help when he’s home but the reality is he has to work a ton of hours just for us to afford to get by, so I’m mostly on my own through the day.
So onto the advice I’m seeking- we have enough saved to cover down payment, moving costs, etc, and have been approved for a mortgage, but not enough to cover an emergency such as a new roof, plumbing issues, etc. We don’t want to make a stupid financial decision, but I also feel like I’m going crazy not being able to do much for the majority of the day besides be in our room or walk the girls up and down the driveway. Getting out of the house is nearly impossible when I’m alone with two babies. We really don’t want to rent and be back in the same place we were before, living paycheck to paycheck and unable to save. So what I’m wondering…do we take the leap, buy a house, and risk having to put a large expense on our credit cards/take out a loan? Or do we try to stick it out until we have a few thousand more saved? I know we’re young and one day we’ll look back and this will just be a speck of time in the grand scheme of things, but at the moment we fell like we’re going crazy. Thanks for sticking around if you read this whole thing lol.