I really do feel for you, I know exactly you're feeling right now.
1st thing, people with severe ME can't shower every day. Being severe means being unable to attend to your personal care by yourself. Someone with severe ME would need someone to take them to the shower and wash them. It sounds like your friend is misinformed.
I try to bath every 2.5 days. I have sebhorric dermatitis (which I can't spell) my face and hair are incredibly oily so I look awful on day 2, these days I scrape it up into a messy bun. I did have it cut short in year two of ME because it was such a struggle, but I soon realised that it takes more effort to style short hair than long, so I've let it grow again and I never bother styling it or even drying it. I just comb out the knots and let it dry naturally. I use combs or clips to make it presentable if I'm going out.
Funnily enough the best thing I did for my self esteem after getting ME was to get rid of many of the mirrors I had about the home. It's a very liberating thing to do.
Can I reassure you if you rest and ignore the housework you will have reserves of energy left to do the quality time things with your kids like going to their school events and on holidays. These are the things they will remember - they will not thank you for a clean house - they will thank you for prioritising the things that matter to them.
The kids are old enough to make their own breakfast and take a turn a week at cooking dinner - you are being a good mum by teaching them self reliance and domestic science!
At the weekends get the kids to help you cook a huge pot of something you can freeze portions of, like chilli, spag bol or curry (You can sit at a table to chop vegetables btw - 6 years it took me to realise this!)
Sit down to fold washing, make piles next to you and get the kids to put their own things away, make and change their own beds. They are old enough and perfectly capable. If their rooms are a mess, shut the doors!
I insist on tidiness in communal rooms and get my husband to help clean once a week, I do surfaces he does the floor. He changes the bed, I change the pillow covers. It's quite nice doing these things together actually, it brings us closer.
You can sleep during the day, you can sleep from when the kids go to school until they get back if you want, you need rest. You can sit on the sofa and watch tv all day. You are ill, you are allowed to sleep and rest.
I stopped driving 6 months in, I just wasn't safe. People weren't happy about it, but they weren't the ones having all the near misses. If you don't feel confident driving you don't have to do it, you're an autonomous adult - do what feels right for you.
If you can't get out for family days out try a wheelchair, I know it's a hurdle, but if it means you can go out with them, then it's worth it.
Basically, you can't fight ME, it will win every time. So let go of the things that don't matter save your energy for the things that are truly important.