I'm a WAHM, and I do most of the household stuff. DH isn't interested or proactive, but if I ask him to do stuff, he'll do it. And there are a couple of things that are his jobs and he takes ownership of on a weekly basis. Those are 1) taking out the trash every morning and 2) taking the trash to the curb on trash night. He also washes dishes about 50% of the time, and I've taught him the Flylady way of shining sinks.
Here's something I think might work if you're trying to get help without nagging. Some people post a list of daily chores on the fridge, in a plastic page protector with a dry erase marker. The list is up there for everyone to see - that way there's no "I didn't know that needed doing" excuse. Whoever does that chore for the day checks it off.
Here's our chore list, if you need a sample. Warning: it's very detailed, because I needed all these reminders when I first started Flylady.Morning
squeegee & wipe down shower walls
swish & swipe toilet and bathroom vanity
put on a load of laundry
toss last night's dry dishcloth & towel in the dirty clothes; replace with clean ones
look through calendar
thaw something for supper
take DD to school
5-minute room rescue (basically, 5 minutes of straightening in your messiest room)
15-minute declutter or 2-minute Hot Spot Drill (depending on how much time I have)Evening
Family pick-up time
straighten up kitchen
wipe countertops & stove
Hang dirty dishcloth & dishtowel to dry
2 minutes straightening hot spots
look at tomorrow's calendar
wash face/brush teeth
lay out clothes for tomorrow
put DD to bed
Regarding DHs being slobs - I know exactly what you mean. I've been married for almost 10 years, and I still don't know if DH literally doesn't see dirt/clutter, or if he just doesn't care enough to lean down and pick it up. I've given up caring. My mantra now is, if it bothers me, I fix it. I don't keep score or count how long it takes DH to do it, because that makes me feel like a martyr. Most of these things take about two seconds to rectify. If I sat and stewed about it, I'd waste much more precious time and energy than it takes to just do it myself.
Your DH sounds like a keeper.