I sometimes see the dish drainers you describe at Goodwill. The problem with going in to look for something you actually need is that you'll find a lot of nifty stuff that is cheap and functional but adds to the clutter. Occasionally you'll find things you didn't know you needed, such as the case with the $20 tall narrow table in my kitchen that serves as an island - what did I ever do without it?

Years ago I found an odd little drainer at the Container Store that of course they don't carry any more. It's a hard rubber surface with a slot for silverware, a few inches of area for setting cups or large stuff that doesn't fit into the final few dish slots. It sits on a towel, and every so often I stuff it into the dishwasher to freshen it up. I took a look online just now - the designs are innovative and the prices eye-watering.