Can I get a replacement big-game tag if my animal is diseased?

Yes, but you will need to contact your local Fish and Wildlife Officer Wildlife Biologist, or take the entire animal to a WDFW Regional Office in order to do so. Expect to surrender the antlers or horns of your big-game animal to WDFW, and to have to dispose of the carcass.
Once approved for replacement, you will need to contact WDFW’s Licensing Division in Olympia so your original tag can be voided. That can be done immediately, allowing you to obtain a replacement tag from any license dealer at no charge other than a dealer processing fee.