Let me start out by saying that you're right, it shouldn't be this way. There are tons of other mamas in this situation with you, so you aren't alone. BUT they key to changing it lies within YOU-- not them.
I'll add my flylady.net endorsement. It isn't a magic bullet, but FLY stands for Finally Loving Yourself... and that is what she works on (along with getting the house cleaned up). You do get a lot of email from it, but a lot of times someone will write in and share how they got their husband or kids to help... she is always saying "our kids are watching and learning" and reminding you that you do shape their attitudes about housecleaning, etc. My little one is too small for housefairy, but when he learns to walk, he's getting a feather duster, lol.
I think you really need to work on "fixing" your self before you try to "fix" the hubby and kids. Understand what you are willing to do, cheerfully and as a blessing to your self and your family, and what you are not willing to do, and then learn to live cheerfully with what doesn't get done. Try to see that your husband not helping around the house is not the defining characteristic of your marriage. Not that you have to go all Stepford, but embrace the fact that there are somethings you can control (your attitude, your tone of voice) and somethings you can't (his attitude, his upbringing). Once you master that, things will change. Also, men love to fix problems, but hate nagging. If you can present certain things to him as your problems you need help with (I'm having a rough week, and I'm so stressed out, do you think you could help me lift this heavy thing, clean up the kitchen Wednesday night, what ever it is). That definitely works around here.
The kids will respond to a combination of following a good example (a cheerful good example, not a martyr good example), rewards, and consequences. I'd set up a sticker chart (ok, maybe something like that for the 13 yo, but not stickers) for agreed-upon, age-appropriate chores. Each of your boys is old enough to put laundry in the correct hampers or baskets or whatever you do, and your 13 year old is old enough to wash and put away his own laundry. If they do it for some pre-determined length of time, they get a reward (stay up 30 mintues later Friday night, pick out a movie from Netflix, whatever). If they don't, then you'll get to the laundry when you get a chance, and if they don't have something clean to wear to school, then that is the consequence.
This isn't going to change all at once, but it will change. You will change. Things will get better, I promise! :-) Give yourself credit for all you do, and don't let anyone (even yourself) take that away from you.