Ok, whether your normal role is a sahm, right now, you are on the equivalent of maternity leave, which changes things. Plus you are breastfeeding which is an intensive role which can't be shared.
Firstly, have you got any help near you? Family or friends, who might do a bit of cooking, cleaning, or take the older kids out?
Is the eldest in nursery? If not, when will they get their free hours? Could you afford to put the eldest 2 in nursery even for one session a week? This would let you have some time with just the baby, to catch up on sleep, etc.
I would say that routine is key here, and you are such early days, it's hard to see one right now, but it won't take long. Personally, I would suggest you both get up at the same time if the youngest is awake. Sharing breakfast together can be a good way to bond as a family, which is important with a new addition! Then a quick tidy up together, DH watches kids while you get dressed, and you begin your individual days.
I would then force an activity every single day. Summer is starting which means local leisure centres and children's centres will put on activities. It's hard but I found that getting out and letting the toddler(s) burn off energy. Allows you to have a cup of tea and snuggle baby, plus maybe they will nap in buggy or car seat. Also stops house getting messy.
Home, have lunch. Trap kids in high chairs or booster seats while you make food, and if baby won't lie down and watch, then maybe a sling would help? Get them used to staying in their seats while you eat and clean up. After lunch, settle older kids down for nap or telly time, so they get used to this time of day being quieter. Baby will before you know it be napping more regularly too.
Is it possible to contain kids stuff into one place?
Then I would suggest sharing what you can in the evenings. When I was breastfeeding, DH cooked dinner almost every night as I was feeding her all evening. We both just knew that was how it was and it wasn't permanent. Now she is 9 months I have my evenings back, I cook, and the house is tidier. It does get easier.
I would strongly suggest that on a weekend, you each get as big a lie in as you can. E.g. On Saturday, you sleep until 12, with DH doing evyerhting child related and just bringing baby to you for milk as and when. Then he gets the same on Sunday. One decent sleep a week makes such a difference. It's also good for DH to see that coping with 3 doesn't leave much time for cleaning and cooking.
Cleaner if you can afford it.
Online shopping (can do while breastfeeding in the evenings) weekly with top ups where needed but easy to do with kids in tow compared to a big shop.
Depending on how tough you are finding it, home start might be able to help - they can provide volunteers to come and play with kids to give you a break.
Don't be afraid to use the TV at this point. There is no shame in it.
Remember baby will start to understand night from day soon.
Is co sleeping an option?
Could you take over household admin from DH? Just because it's a job which can often be done on a computer and therefore you could do it in that 'wasted' time while cluster feeding.
If you can get any help in the week, even if they only play with and cuddle the kids, take the opportunity to sleep, or batch cook for the freezer etc. If people offer help, don't hesitate to ask them to hoover, or cook a meal, or babysit.
And post here. It's early days and 2 to 3 is a bigger change than 1-2. It sounds like you are both doing what you can, but maybe DH needs more understanding, and I think you need to be given more compassion!
Agree that it's worth reminding yourself how great you are doing, but also how great DH is doing, and to tell him so. You must love each other, so appeal to his understanding of how hard you are finding it. As most new mums do!
Keep posting, you have support here x