I think you need to start at the beginning.
How old are your kids? Because honestly, when you have just young kids it is HARD, really hard to be a stay at home parent. You spend all your waking hours with people that need-need-need-need and you feel depleted. That isn't sinful, that isn't selfish.
Right now being a mother of 6 is far and away easier than what it was like to be the mother of 1 or even 2 children that were young. Right now we have 12.5, almost 9, 7, almost 5, 2, and 1 year old children. That sounds like a lot of work, and in a way it is. The wonderful part is that I don't spend my day just wiping bottoms, wiping noses, and doing housework. I can have engaging conversations while doing all those things!
Cultivate hobbies and dedicate time to them in a structured way. Have time for you, and only you each day. Everyone has different ways that they recharge, find out what yours is and start doing it. You won't get to spend as much time as you might like, but don't neglect it just because you can't do it very long. My husband enjoys throwing knives in the backyard, spending 15 minutes a day doing that helps him tremendously. I enjoy knitting, and do it in the evening after the kids fall asleep. I find that losing a little sleep to knit is more beneficial than getting a solid night of sleep.
What you are feeling isn't necessarily sinful, or even awful. Be honest with your family that you need a little space to reset every day. Being the stay-at-home parent is a 24/7 job. Not to be sexist, but it isn't a role that is normally done by men either. So you aren't going to have much support. If stay-at-home moms don't fell real supported, how much less supported are stay-at-home dads going to feel? Don't fall into a trap of trying to prove your worth and have everything perfect.
Step back a bit, take a deep breath, and repeat "this can't last forever" while sipping a rum and coke. Childhood is fleeting, try to look past the mess a bit more so you can enjoy it.
The greatest and most important parts of my days are the conversations that are started with my kids. My husband will come home to a house that is a mess often. But he also comes home to children that have asked deep questions that were answered in the best way possible the moment that they were asked. The dishes can wait when my 9 year old wants to understand why God allows suffering to exist. The laundry can wait when I need to calm the fears and raging hormones of a tween. I would rather have happy children than a perfect house.