A couple of things stand out in your posts, you say you don't want to work full time, but want him to help more, ....realistically he only has the weekends.
You can only just manage financially but want him to prioritise family time over work and still not work full time yourself...how can that work?
You also are not willing to move to a cheaper area to ease the pressure.
You hate the drudgery but still had 3 kids! of course there is a lot of drudgery if you have 3 kids.
Its simple either work full time and pay for domestic help to avoid the drudgery, and share any admin on the weekends, or stop complaining and make it work.
My DH has never ever helped at all evenings or weekends with anything, no housework, DIY, mowing the lawn, paying bills, xmas, birthday, nothing...nor got up with sick Dcs.. ever.
But he leaves at 6.30am and returns at 7pm or later in an immensley stressful job.He has travelled most of the past 3 weeks.
I have worked PT (partly from home) since Dcs, and done everything else from decorating to taking Dcs everywhere, to me it is a fair division of labour.
Of course l hate the drudgery, but he hates travelling to various time zones and being in hotels.
I hate unblocking the sink, he hates getting up at 5.45am daily.
I get to have the odd lunch with a friend in the day or go shopping occasionally , he gets to have work lunches and functions, and drinks after work.
If I was really fed up l could work more and pay for more help, that is my choice as it is yours.
As long as you both have the same amount of leisure time that is all that matters.
Stop hating your Dh he works hard to support your family, take control of your time to suit you more, and if there is truly still an imbalance talk to him about dividing time spent on things at the weekend.