I sooooooooooooooo feel you. My kids are about the same age as you, and some days they drive me crazy. I bring some of this on myself, by not parking them in front of the TV. But believe me, I don't blame anyone who does: everyone needs a break and they're probably less frazzled than I am. At the end of the day, being smug because I'm TV-free doesn't do me a heck of a lot of good if I'm really wanting to scream "Get away from me!"
Part of my problem is that, unlike literally every other family I know, we have NO family nearby. So I get no free breaks, whereas everyone I know gets date nights and to drop their kids off and go our to coffee or for a run or a pedicure. They go away for the weekend while their kids stay with grandma and grandpa. I can't even imagine how nice that must be. If you have family nearby who is willing to look after your kids, I would absolutely take advantage of that. I'm not sure why you think it's "dumping." The 1950s sit-com family, where mom does 100% of the child-rearing (with dad occasionally pitching in) is not the way that children have been raised pretty much anywhere over the history of the world. IMO it's good and healthy for children to have a range of role models and instructors. Obviously, if the family in question is toxic or dangerous I wouldn't do this: but I don't consider slight deviations from my parenting standards (junk food, TV with some limits) toxic or dangerous. My daughter well understands different homes have different rules.
Despite money being very tight, I finally broke down and got a one morning a week sitter this summer, and it is a godsend. I really just use that time to run errands, but it now takes me about 3 hours to run all the errands that I used to do in a week or two. Lugging two small children everywhere, particularly to places where you have to stand in line and wait for your number to be called, or where you have to try on clothes, or where basically ANY sort of quiet and patience is called for is just asking for trouble. Don't get me wrong, sometimes my children are absolute angels. But it's a crap shoot, and the stress of worrying about how long I have before someone starts screaming makes dull tasks downright awful. It's also just really nice to have a little bit of time sitting in my car, cranking my music, and being able to run into my favorite bagel store without unhooking two kids out of carseats and carrying the non-walking one in and making sure the walking one doesn't run into traffic.
An even cheaper option than a sitter, if money is tight for you, is a mother's helper. Around here that's a 9-12 year old neighbor girl who only makes about $2-$5 an hour (as opposed to sitters who make $10-$15 an hour). They're too young for you to be able to leave them alone with the kids, but they'll play with your kids and occupy them while you get some things done elsewhere in the house, or even just relax with a book. Kids that age are often really good with kids: they're young enough that they're still great at playing and they love getting down on the floor and playing games and being silly.
I also joined the Y, which has a wonderful childcare room. I'm currently having some pregnancy issues that keep me from working out (except swimming), but you know what? Who cares! It's air conditioned and I need some quiet time! I still drop them off for an hour or two most every morning and if I don't go swimming, I sit in the lobby with a book. It took them a while to get used to it, but there is a giant selection of toys and activities and they do arts and craft projects and I'm thrilled to let someone else clean up after their mess for an hour or two a day. Again, although this is an expense that we probably should cut to be fiscally responsible right now (and I did cancel my membership... for two days and then I went and begged for it back!) I really need that time for my sanity.
I hope you've also considered putting the 3 year old in preschool this year. That will also give you some quality time with your 1 year old. I never realized how easy one child was until I had two! My daughter went to a toddler program last year and will go to a Montessori program this year, and she just loves school. And I really like outsourcing keeping her occupied, which is a tough task. And I don't think she's even particularly high needs: I just think that 2 and 3 (and 4 and 5) year olds are very social creatures, and very rambunctious and energetic and have short attention spans and need a lot of structure and activities to be happy.
Your final issue is your DH. DH and I trade weekend mornings: one day I get to sleep as late as I want while he's up with the kids, and the other one he gets to sleep in. All bets are off if the kids are having an especially *ahem* "difficult" morning. I also don't do more than two poopy diapers on a weekend before deciding it's his turn and sending in said poopy child to jump on our bed until he changes them. I change enough during the week, that on the weekend I'm not putting up with getting stuck with all the poop too.
But seriously, it's DH's own fault if he stays up late and is tired the next day. Guess what? There are things that I'd love to stay up all night doing. When he's at work and you're at home, obviously you're responsible for 100% of childcare and housework. The second he gets home, the workload is 50% on each of you. Why does his job get to end, while yours is 24/7? That's just not how having young children works. And that's not to say that you have to have some tit for tat arrangement where neither of you ever get a break... just the opposite, actually. He needs cool off time from the stresses of his week, but so do you.
((((Hugs)))) to you. I'm really sorry you're so stressed, but it does happen to all of us. Just remember that a Martyr Mommy is not a Happy Mommy. You need to take care of yourself so that you can take care of your kids.