You sound like depression has moved in to your body. See if there is a low-cost or sliding scale mental health clinic in your area. Call your town's department of Family Services - they may be able to to help, or refer you.
Since you don't own your home, the landlord is responsible for pest control. Report that you have roaches and they must send an exterminator. No matter how much you clean, you cannot get rid of them yourself once they are there. They are in the walls and come out at night. But they are a health hazard so whoever owns your duplex (landlord or city housing, whoever) is responsible. Your dirty house didn't bring in the roaches - they came from somewhere else. It's not a question of blaming the other neighbors. The roaches are there and the source doesn't matter and cannot be traced.
Same for the sink - they can't refuse you because you place is cluttered.
Take all your dirty clothes to a big laundromat. Throw everything in a few giant washers and get it done. Sometimes kids like watching the big dryers go around. Otherwise take some games and pass the time, let them help transfer from washer to dryer and put the quarters in. They can learn to fold their own things too, or the little ones can sort socks. Make a game of it. Maybe you can find a washer or dryer at a moving sale - sometimes people don't like to move big items, or they can't fit their current machines in their new place. My daughter got a fantastic new (well, used) washer dryer, one of the front loading washers, for $200 in just that way.
Cleaning - pick ONE room or ONE type of problem. For example, choose to collect all the magazines & newspapers, and recycle them. Or choose all the toys. Or just choose the living room. Next week, choose the kitchen. Give the kids an empty wastebasket or a box, and have them pick up all of one category - all the books, all the trucks, whatever. Many hands make light work. Or tackle JUST the kitchen. Get rid of stale or expired food, get rid of the empty or clogged up bottles under the sink, clean off the counters, organize the cupboards. Start small but finish the one small job.
Use simple cleaning products like baking soda for the kitchen and bathroom counters/sinks. Cheap, not hazardous, no vapors. It does a great job. It's good for the inside of the refrigerator too - borrow a big cooler, put all your perishables in it, remove the fridge shelves and wash them (or rinse them in the shower and leave them in the tub to drip dry while you clean the fridge door and insides with baking soda & water. If you have little kids, do not let them get in the empty refrigerator!! By the time the fridge is clean, the shelves will be dry and you can put them back. Then put the food back in but organize it -- all the dairy together, all the fruits & veggies in the drawers, all the condiments like ketchup & mustard & mayo all together, and so on. Get rid of anything stuck in the way back that spoiled a while ago - we ALL have something like that lurking in the back corner and it doesn't mean you are a bad mom!!!
Just go forward one day at a time and don't look back on how organized you used to be. Just do a little bit.