Ours are a little less stressful but feel similar here's what I do (may help, may not hah) Also work FT out of the home! 2 kids aged 9 and 5 and no family support.
Once a Month each DH and I get together with friends, separately but this could be with the smalls in the park, or a girls night out or just a lunch and coffee somewhere. can be form 1 hour - to a whole evening but it is prioritised for each of us.
We do all our own cleaning but bathrooms are done midweek (2 bathrooms, 1 has an evening each) and vacuuming is also only done upstairs once a week on another weeknight (usually while 5 is in the bath) DH works from home and vacuums downstairs daily (thanks to having 2 cats) Also do a 'closing shift' on the kitchen and lounge every night.
Ours does Saturday morning training and Sunday morning swim lessons - in the same places thankfully so we actually get a nice coffee and cake together while the training happens and take turns to do the swim lesson duty to give the other a lie in.
We do a wash load a day, every day, sometimes 2 (washer on am, go to work, come home hang out and then another load in if needed) this seems to ease up on the backlog through the weekend. Kids are responsible for putting away after it's folded and on their beds (even the 5 year old) and if it's not in a basket it doesn't get washed.
Ours bicker too - I've learnt that as long as its not harming them to leave them to it.
I was doing a big Lidl shop on a weekend, but I now do this one a Thursday night after work, its open until 10 round here and only a 10 min drive away so DH will do bedtime and I'll go shopping.
I bake a cake and granola bars every Sunday afternoon, then also cookies or gingerbread men get 5 involved and he now knows how to make most wit barely takes any time at all.
My Dh plays golf, he will often try to do this on a Friday afternoon rather than a weekend day, but if it falls that way I get the kids out for a family swim or something in the garden - outside for the win!
We too have an open plan downstairs and it's a bloody nightmare!
so yeah, not much different to yours - and we don't spend much alone time together but hoping it'll pass and we will miss these days when they're gone!