Is a cheap dishwasher that actually dries plastic even sold in the US?

What you're describing is real, and it comes down to how the two markets build machines differently. European dishwashers mostly use condensation drying: a stainless tub, hot water sloshing around, moisture condenses on the cooler steel walls and drains away, then the door pops open to let the rest vent out. Cheap machines there get that as standard. Over here the entry and mid-level machines lean on a heating element in the floor to bake everything dry, which is why your plastics end up holding puddles and why the element worries you. Your instinct about it is fine, though. It's ugly on energy and it will melt a plastic spatula left in the wrong spot, but it's not going to ruin your Tupperware on the top rack if you use heated dry sparingly or turn it off entirely.

The annoying part is that the door-pop style machines do exist on this side of the Atlantic, they're just not at the price point you'd expect based on Europe. They sit a couple tiers up from builder-grade, so your $400 budget lands you exactly the flimsy stuff you were looking at. If you watch for sales around holidays you can sometimes catch one in the five-something range, which still stings compared to what your parents paid. A couple of things that helped me in the meantime. Rinse aid makes a bigger difference than people think with condensation or residual-heat drying, and most builder-grade machines ship with the dispenser empty. Fill it and turn it up a notch. Also, plastics genuinely can't dry like ceramics can, since they hold heat poorly and the water just sits there, so angling them on the top rack helps a lot. One more thing to check on whatever you buy: some US machines skip the internal heater entirely and expect your hot water line to feed them. That's a trap in a house where the water heater is far from the kitchen, because the first fill is lukewarm and the wash suffers. If you go shopping, ask wether the machine heats its own water. Any machine that does, plus a stainless tub, is the closest thing to what you had back home.

This explains so much about my machine. I always thought the wet plastics were my fault for stuffing the top rack too full, and here I've been opening the door a crack when the cycle finishes to let the steam out, felt kind of dumb doing it but it does help. Never even knew there was a little door thingy inside for rinse aid untill now, going to go look for mine tonight. My machine is from the builder so I have no clue what it is, is there some sticker or label inside the door that tells you if it heats the water itself?

Same story here, moved over from Portugal a few years back and the wet plastics drove me crazy at first. The thing that actually made the difference for me was dealing with the water itself. Our town has stupid hard water and the dishes were coming out with a film on top of the puddles. Put a softener on the whole house and suddenly the glassware looks like the commercials. Might be worth checking how hard the water is where you are in Ohio, some of those towns are brutal. Doesn't fix the drying issue obviously, but it makes the machine you allready have wash a lot better.