I never wash wood handled pans or utensils in the dishwasher. I hope that, if you do, you are treating them often with foodgrade moneral oil, but that is likely what's making them feel " icky."
Wood , imo, should not be washed in a machine. Trying to convince my daughter of the same...

I have put my cheap wooden spoons, wood handled cooking spoons and forks and measuring spoons in the machine for decades. A lovely wood salad bowl I would hand wash. I am looking for a product that works better than LemmeShine, which still can leave a haze on my stainless steel tableware.