Work as a team. Suggest a time to get together to assess priorities - what absolutely has to get done, what probably should be done, and what can slide. Get agreement on each other's priorities. Then divvy up the essential tasks on a schedule that works - let each person pick the things they prefer to do and avoid the things they hate (if that works out!). Be reasonable about stuff that's not essential - for example, maybe the beds don't have to be made all the time. Figure out if there are some things that are better if you do them together - folding sheets, changing linens, etc. Folding laundry can be done while you both sit and watch TV after the kids go to bed. Sometimes vacuuming & dusting are done productively together - one person straightens & moves the furniture while the other vacuums, and so on. Bathrooms are usually a one-person job, but perhaps that one person can do all the toilets in sequence and wipe down all the sinks, once you're in the groove. My husband and I unload the dishwasher together - each stands on one side and unloads/puts away the stuff we're closest to. Works great. You could also try cooking together on the weekend - make meals for the week. Experts are saying it's important for kids to be involved in this too - they tend to eat healthier foods and the family relies less on processed, quick meals. Then try to at least have family meals on the weekends so the kids get to eat with you and Dad at least 2 days a week. Maybe you can set up an assembly line for the week's school lunches - if you have 2 kids and need 10 sandwiches, make them all on Sunday, and freeze those for later in the week. I don't know how old the kids are, but they can start taking the responsibility for some of their needs, like grabbing a sandwich from the freezer (it will defrost well before lunchtime), a drink, and a piece of fruit, and sticking it in their own lunch boxes.
You say you are fine with him being out 2 evenings, but are you? When are you out doing something by and for yourself? When is YOUR time? I'd schedule that in - maybe it's a Saturday afternoon or another weeknight - but if you're out doing something fun or rewarding, and he's in charge, he'll connect much more with what it takes to be the sole adult in charge of everything.
It's important for husbands to realize that they aren't "HELPING" with the kids - they are PARENTING them! He needs to be involved. That said, it's important that you "let" him do things the way he wants to if he's in charge, and that he's not just responsible for doing things your way. You have to relinquish a little control. If you don't have a family calendar, put one up. List everyone's activities - soccer games, parent nights, whatever, and then initial who is going to be in charge of getting the child prepared and delivered to that activity. That means the drive, the equipment, the snacks, whatever.
Kids can also set the table and sort their own socks after the laundry is done (also while watching TV!). And they can put their dirties in the hamper or bring a basket down to the washing machine. I kept a basket in my child's closet, and his stuff went right in there. Helped keep his room clean and made stuff ready for the washing machine.
If you can find some shortcuts, and then make the rest of it a family project, you may be able to find some great accommodations and time savers.