You def need to try A Slob Comes Clean! She's amazing and really gets into the nitty gritty of why all the "just do this!" doesn't work. I can't focus on podcasts unless my hands and eyes are busy, so I put it on while I'm cleaning or tidying up or folding clothes to put away etc. It's become sort of linked with cleaning in my mind which helps as well.
Is it possible for you to get a dishwasher? I kept an eye out and bought a second hand one on ebay years ago cash on collection. I had to swap over the water connections from the washing machine every time I wanted to do a wash as I had no splitter but it was worth it anyway. It made a huge difference. I was just not keeping up with things without that. Dishes is actually the most important thing (according to Dana) and I'd agree with her that it makes the most difference, because old food hanging around REALLY makes the place look untidy, much more so than anything else (except perhaps actual rubbish - so I'd combine - dishes and bins - these need keeping up with if you can't manage anything else, prioritise those). They also tend to get in the way and make it difficult to just do things - if you can't use the dining table for homework or colouring or crafts, if you can't easily cook in the kitchen without doing a whole clean up operation first, it makes it hard for anything to happen in the right places. And when you don't have an accessible dining table and cooking space, it makes it hard for mealtimes to be a set activity that happen at a set time, which means (IME) food gets consumed elsewhere while you're not fully paying attention and small hands manage to transfer food mess to a lot of other things in the house which means there is a constant depressing layer of just grime on everything and it feels really hard to keep on top of this.
Any kids who are old enough, ROPE THEM IN to washing up and keeping on top of bin emptying. This is as simple as getting them to take their plates out to the kitchen, drying up (doesn't matter if they do it badly, it's just building an expectation/habit), taking a closed bin bag to the outside bin if they aren't old enough to actually deal with the emptying part yet.
I also really HATE doing dishes by hand because of all the sensory factors around it - smells and the water being wet and it being hot and then getting cold too quickly, slimy things, bits coming off and floating around in the water, residual grease, smelly sponges/cloths, the fact you stack things up to dry and they are just there forever - gaaaaaaah.
My most ADHD friendly washing up routine was to stack everything ready to wash on the left side of the sink and first boil a kettle and fill the sink with literally boiling water. Wearing gloves and/or using a long handled scrubbing brush, dunk everything in the water one by one and give it a very quick once over rinse with the boiling water. Stack again on the left. Drain the water and fill with normal washing-up temperature hot, soapy water and wash things up as normal. If the water changes colour, looks greasy or the bubbles disappear, drain all the water and start fresh with new water. But you get a lot longer out of the water by doing the boiling rinse thing first.
Also, I bought washable dish cloths and I put them into a basket by the washing machine immediately after use rather than letting sponges sit around and breed bacteria, which I find disgusting. When the basket gets full, it prompts me to collect bedding and towels and do a 60 degree wash.
Dana's theory about "Dishes Math" is interesting too - she explains that when you leave dishes piling up foe days and then do them, it takes ages, probably over an hour, so you get into a mindset of thinking "Oh it's not worth doing them yet - I'll wait until I have enough that it makes sense" because you think if 3 days' worth of dishes takes 90 minutes, 1 day's worth of dishes is going to take half an hour. But realistically one day's worth might actually take more like 10 minutes because the longer you leave it the more the time multiplies. It doesn't make logical sense, except that it does - it's to do with stains being more set and dried on, as well as having to re-run water multiple times, as well as not having drying space for everything at once, as well as having more stuff to wash because you've always run out of things so you get more things instead of re-using what you already have.
For other tasks I have been using an app called Tody which I really love. It sets out what needs doing by order of urgency and I find this helpful because I just don't really get a sense by looking around the house of which area is most important or where to start. Partially because everywhere can be chaotic but also partially because I don't really know what I'm looking for. So the app is helpful. It took me 2-3 attempts to really stick with it but it's working well for me now. The key was getting the amount of tasks right and starting with a small amount. I kept setting it up with everything I could possibly think of and that was unhelpful. The thing that made it finally work was giving myself a rule "if the task stays on the list too long and is stressing me out, just delete it". Because yeah sure maybe cleaning the bath IS the most overdue task in my house, but if I haven't done it in a year and nobody has died, nothing bad is going to happen if I ignore it for another 6+ months. Build on achievable successes first rather than smacking yourself in the face with your failures every day, that helps nobody!