My ex was a lot like that. He was on disability (back injury), but he was not so badly disabled that he couldn't do anything. I worked full-time, and would often be greeted when I got home with "Hi, I'm starved. How soon is dinner?" He didn't clean because he said that vacuuming, sweeping, mopping, or bending over to clean bathrooms hurt his back. He spent most of his days in his recliner watching tv. He would call me at work to ask what there was to eat - all he had to do was look in the fridge and pantry. He wouldn't even pick up his own clutter - there was stuff stacked on every flat surface in the house, and he would have a fit if I threw out a stack of old newspapers if he hadn't finished reading them. I told him that after this long, whatever was in them wasn't news any longer.
He would write checks for bills, not record them, and not tell me he had written them. I would use the debit card at the supermarket, and then,of course, the checks would bounce. We were constantly overdrawn.
He couldn't clean the house without hurting his back, but he was always ready for sex, and couldn't understand why I wasn't.
The man I'm married to now isn't like that, because we dealt with all that before we decided to live together. I work full time, he's a full time grad student and teaches part time. We split the cleaning chores, we take turns cooking. If I cook, he cleans up the kitchen and vice versa. We maintain separate bank accounts and split the bill paying. He covers the rent, gas for his car, and the three cell phones, I cover car, health, and renter's insurance, gas for my car, and utilities. We keep a running grocery list, and add items to it as we see that we're getting low. We take turns doing the marketing, and we often do it together. Everyone in our household is picky about how their laundry is done, so he does his, my daughter does hers.
As a reult of my expereince with my ex, I have little patience for that kind of behavior. If he has time to play video games, he has time to help with the upkeep of the house. He lives there too, he creates part of the mess, and child care does not automatically default to the one with the most estrogen. HE knew when he married you that your 5-year-old was part of the package, and he helped make the 1-year-old, so he can damn well help take care of him. He needs to nad up and start acting like a man and not a spoiled little boy. You're his partner, not his maid. Not to mention the fact that he is setting a terrible example for the boys regarding how to behave and how to treat the women who will come into their lives later. And if he does not respect you, pretty soon your sons won't either.
If he won't listen when you ask for help, then you have to get the message across another way. Stop picking up behind him. Tell him from now on, you will clean up after yourself and the children, but not him. The 5-year-old is old enough to be able to pick up and put away his own toys. I'd start drilling him on that if you haven't already.
Stop cooking if he won't help in the kitchen. Buy bread and cold cuts and make sandwiches for dinner every night for a while.