I'm NOT an expert...
...but I think you are hearing the beginning of your rear end's side yokes wearing out.
Welcome to Corvette Action Center, Silver1976. Hopefully someone will have a better idea than my own...
My own differential would make groaning, moaning, grinding and binding noises when first rolling from a stop and also going slow around any turns from a stop or slow speed. I am sure the noise began quietly at first, but then progressed to what I remember (listed above).
I would recommend you take your vette to a vette shop, mention the symptoms you are experiencing and ask them to check the side yokes.
Due to wishful thinking and denying the inevitable, my side yokes became so bad we needed to replace the entire rear end. Maybe if your vette is experiencing the same problem mine did, you could have the side yokes replaced without having the expense of replacing the entire rear end, like I had.
When working properly, the drivetrain should not make ANY noises. Mine has been working quietly for the 9 years since the rear end change.
Heidi