Not sure what their parents' wealth has to do with it. Are you saying they were raised with servants and a sense of entitlement? Then they aren't the problem, their father and mother are. Their father has to ask them to do stuff? Then he's aware of it. But if they just throw things in the sink, have they been specifically told by him what is required? Or is he just letting them do the bare minimum to appease you?
If he wants them to do more, then he has to demand it, or he has to apply consequences for their defiance. Why, for example, are you still bringing all the food to the table while they are eating? Why isn't everyone given a few dishes to bring in together? What do you mean by "express gratitude"? Do you want them to thank you? If everyone helped, then the thanks would be mutual. Or are you suggesting a before-meals blessing? If the latter, is that their religious tradition?
What do you do when they jump up and leave? Do you critique them to their father? I think I might just sit quietly and finish my dinner in a leisurely way, and then the next night I wouldn't be cooking since it's not appreciated and you apparently aren't cooking what they want/demand since they are bolting from the table before you even sit down. So I wouldn't keep going back for more punishment. I'd tell them to fix what they like as you have put plenty of things in the fridge, and if there's anything else they want, they can just put it on the grocery list or go pick it up themselves.
I suggest that your problem is with your husband's parenting. So work with him. Or, if you are trying to impress them and be accepted by them (which I completely understand - I'm a stepmother of 2 girls!), realize that this isn't working.
I don't understand Julie S.'s indignant comments about not being able to afford the right dishwasher. I don't think that's your point. But I do think that you labeling them in a certain way, as if their economic status requires a level of manners that one shouldn't expect of poor kids, really sells out the kids who were taught skills regardless of their parents' incomes.
It's also possible that this is a form of rebellion - not just as teens but as kids who have a new woman in their lives and their father's life. If they know better and just aren't showing you, then this is defiance. That's a whole different problem than poor breeding. When a parent marries, things change, and that's hard for kids. If they are as poorly prepared for this marriage as they are for life, the fault lies with their father. He's handicapped them for their adulthood.
I suggest a good sit-down with him, and perhaps counseling, to say, "This isn't working, I'm getting frustrated, and that's no way to live. So instead of me trying to please them with efforts that aren't appreciated, how about I just bow out of all the domestic stuff and let you handle it, as you did before I came along?" Then go back to filling the fridge only and making your own delicious dinner for 1 (or 2 - for him) and letting them happily choose what they want. Put out paper plates and be done with it. You're not their maid, you're right. If the dishes are there the next morning, it's up to your husband to assign chores or do the dishes himself.