Pre-Rinsing
This will ruin your dishes, as the detergent enzymes and chemicals will "attack" the finishing glaze on your cups, plates and bowls. It makes stuff look dull, and gives everything a rainbow appearing haze (you will also notice unremovable "hard water" deposits on them).
I also suggest reading your dishwasher's manual, to see the temperatures on each cycle. It could be that your "regular" cycle is too cold, and the eggy stuff and sauce is causing everything to sour in the machine. Try using a heavier wash, or something that has more water changes (Heavier Cycle, Scrubber Option, Temperature Boost etc.).
You may also wish to check the filters on the machine, as many newer dishwashers no longer have self-cleaning filters like the machines of yesteryear - so there may be gloop building up on them that is contaminating "clean" rinse water, and causing a stink. If you don't want to handle yucky filters, then I suggest purchasing a dishwasher cleaning product or running an empty cycle with your detergent choice (Hottest, Longest, Heaviest) to get rid of any "buildup" inside the machine.
Try flushing your water heater - even in less than a year, you may have accumulated a ton of mineral sludge and other garbage in there. Maybe even bad bacteria, if your water heater is set lower than around 130º, which could also cause the odd smells in your classes.
I don't mean to be rude with this question, but are you sure it is not bad breath that causes the smell. Sometimes, I think my cup is really smelly and horrible (when its clean) and it turns out I am smelling my own breath - lol. If others are complaining about a similar scent, you can probably rule that out.
And as for detergent, I would recommend Finish, perhaps their powders. I pay AU$18 here, and get almost 100 for 1.5kg of powder, whereas tablets that give you 1/4 of the washes cost up to AU$25. You may wish to purchase some STPP (Phosphates) from an online store, to see if that boosts the washing performance if your detergent is Phosphate free. I know for a fact ours isn't, and with soft water, we get great results. Our detergents (for the dishwasher) still have >30% Phosphates, as indicated on the label, so read the labels when you go shopping.
I hope this information is of help to you. Good luck!