Could you have a family meeting to discuss this and brainstorm solutions? Talk about how you're needing more help keeping the house clean, and you know they don't like to do a lot of the things you ask them to do, but you know if you work together you can all find a solution? Ask them if they can think of ways to make keeping the house clean easier for everyone?
I know that even with my 8 year old, something like cleaning up Legos is overwhelming. The kids really need help with something like cleaning up the Legos, picking up a couple of thousand really tiny pieces is kind of a tedious thing that takes a long time (my kids dump them out onto the floor to be able to find what they need). One thing that makes it easier is to have them dump them onto a blanket, so we can use the blanket to dump them back into the bucket. Even I hate cleaning the Legos. So when my kids resist picking up Legos, the thing to do is to either do it with them or say "okay, if you clean up [something else] I'll do the Legos."
Also, I'm thinking that the way I phrase things matters a lot. "You dumped all these books out, you need to pick them back up" tends to create resistance. It helps to phrase things more like "I see that the books are all over the floor. I'm concerned that they'll get ruined/someone could trip/whatever. Will you please pick them up?/please pick them up/would you be willing to pick them up with me?" (Making a genuine request, which means I will take no for an answer, also leads to my kids' generally being more willing to agree to my requests.) eta that this isn't just using nice words in order to get my kids to do what I want them to do, but about expressing genuine feelings and needs, owning my own feelings, not blaming, and most importantly creating connection between us. That connection is what allows us to give freely, finding more peace and cooperation.
It also helps to take into account not only what's overwhelming, but what chores are really just things they really dislike. I find my kids are a lot more willing to help out when they have choice about how to participate in the care of the house.
With things like clearing off the table after dinner, it took some time to make that routine but making it routine was the key. Now it's habit, after every meal each child clears their spot-putting dishes into the sink or dishwasher. Occasionally they need a reminder, but there's no resistance. It does take patience to make it routine.
It helps to express genuine, clear/specific appreciation for the help my kids give me. "Thank you for cleaning up those Legos, I appreciate your help. Having your help makes keeping the house clean much easier for me/it was much faster with your help/etc."
With things like your son's pull-up, is it possible that even though you think he should remember that he actually doesn't? Is it possible that he doesn't want to carry a wet pull-up (I can imagine my dd really disliking the feel or smell of a heavily wet pull-up)? I find that whenever there's an ongoing struggle over something, it helps to set aside my ideas of what the problem is and ask my child what the problem is--at a calm, later time. "Can we talk about something? I've noticed that you haven't been bringing your pull-up downstairs in the morning. What's up?" And if he says he forgot: "You forget to bring it downstairs. I understand. My concern is that if you leave the pull-up upstairs, your room will get smelly. Can you think of some ways we can solve this problem?" Getting my kids involved in problem-solving and really taking the time to understand their concerns really helps us. You could do the same thing with school work: "I've noticed that lately you've been saying you don't want to do your school work. What's up?" Work at understanding what his concern/reason for refusing the school work is. Maybe he just prefers to do other things, the school work isn't as fun--and if so, maybe there are ways to make school work more fun. Or maybe the reason is something else entirely.
Books that have really helped me communicate and problem-solve better with my kids are Nonviolent Communication, Connection Parenting, Repectful Parents, Respectful Kids, How to Talk So Kids Will Listen and Listen So Kids Will Talk.
I hope your back is feeling better soon.
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