DH and I have been living together for 10 years and only sorted this problem this year, due to the arrival of DC2. V similar to you - DH full-time, me part-time but not prepared to be solely responsible for house, and neither of us remotely houseproud so no driving motivation to solve the problem on either side.
We had limped along for years with him washing (not drying/putting away) the dinner dishes and mowing the lawn in the summer and NOTHING ELSE, him thinking that he was therefore doing his share and me not having the energy to confront either him OR the housework. Result - regular pile-ups of work and regular melt-downs by me.
I told him that when he doesn't clean his own poo off the toilet, or leaves crisp packets on the coffee table, the message that it sends from him to me is "My time is more valuable than yours." This worked for about a week, but things drifted and I just felt resentful.
Finally I said "Being in charge of all the housework makes me extremely unhappy and I can't go on like this. What can we do about it?" Making it clear that it was a shared problem, and I was not going to let it go.
We came up with a housework rota, a very simple one for daily grind stuff, allocating who did the dishes, laundry, tidying and the hoovering. I was really tempted to add in every tiny job e.g. clean shower tiles, polish mirrors, but since these are things I tend to overlook myself, I kept it to the bare minimum of food-related housework, laundry-related housework, cleaning bathrooms & sinks and tidying up. DH had 3 jobs a day (e.g. wash up after dinner, tidy living room at night, put 1 load of darks on to wash) and I had 5.
Later on I added in the weekly & fortnightly jobs on a list at the side for us to tick off. These were the things that we really should have been doing regularly but almost never got to because of the layer of daily crap that needed getting through first - polish furniture, properly dust bookshelf, hoover behind & under things, etc. We both picked something and ticked it off when we had a spare 5 minutes at evenings & weekends.
Also, check out the "Squalor Survivors" website. here There's some great stuff about "demand resistance", which I think we both tend towards (i.e. if I know I need to do something, but it isn't life-threatening, I will resent getting on with it and try to wriggle out of it). I went round the house and took photos of every room in its untouched state and realised that I was living in what the website describes as "level 1 squalor." There have been times in my life pre-kids when I was definitely in level 2. The website advocates micro-bursts of work - do something for 5 minutes now, rather than try and do everything all day once a month.
So try a housework rota but keep it really simple to start off with and build on it. Good luck.