I hate to say it, but I agree w/ CraigFL. The Jet-Dri does make a difference. We've been running ours without for a couple weeks (just kept forgetting to get it at the store) and everything ends up staying drippy wet. Our washer dry cycle is low energy, meaning it doesn't get as hot for as long to really dry things out. It depends on the rinse aid to get the water to drip off.
If yours is an older washer that does high heat for longer time, then those drips of water are going to get evaporated away and leave any minerals left in the water.
Does your showerhead have lots of buildup on it? Thats a good sign of hard water.
As for water softener and landlord. Perhaps you do have a softener for your appartment, however since you do not know about it, you have never kept it running. Ask your landlord if there is one. It is likely to be someplace really inconveninent like a crawl space, or in the basement behind the furnace.
You could put a cartridge filter unit under the sink to filter just the water to the dishwasher. Then you could take that with you whenever you move.
You could get all new kitchenware that is a milky white color. That way the hard water deposits don't show up.