I have one of those little guys at home and went through the exact same thing when I brought him. At first I was worried because his fur looked sticky after every bath, but I looked into it and it turned out that's totally normal. Chinchillas have incredible fur density - like 60 hairs per follicle or something - and when they bathe in sand, all that amount of fur traps particles that can leave a damp and matted feeling if the sand has any residual moisture.
What matters is that the sand is as dry as possible. I keep mine in a sealed jar so it doesn't absorb moisture from the air, and that helps a lot. Still, it's normal for the fur to look a bit dark or compact after the bath while it dries. Leave him in a place without cold air currents and you'll see how in a few hours the fur loosens up and looks fluffy again. If besides being sticky you notice your chinchilla scratching a lot, has dandruff or the smell is strong, then you could check if the sand has mold or if there's a humidity problem in the cage. But if it's just that temporary sticky feeling after the bath, don't worry about it.