I'm not a fan of Royal Purple but read the packaging carefully and make sure it meets the specs outlined in your owner's manual for TOD lubricant (same as tranny). The TOD is pretty easy to drain and refill, so if you think that's your problem it would be an inexpensive experiment to try a couple different lubes, such as Mobil-1, AMSOIL or regular petroleum products.

My IronMan has been producing gear whine under the circumstances you describe, but I know it's just normal gear wear so it doesn't bother me. But if yours is really loud then you've got some other issue likely and it should be identified correctly. If you have a local shop that specializes in 4X4 vehicles then that would be a good choice for a more enlightened opinion. These types of shops usually have experieced mechanics that do all kinds of swaps and transplants which gives them a good deal of practical troubleshooting skill.