I use T-bond II for my segmented Pepper Mills, I concluded the main difference between II & III was the price. If you look on my photobucket site you will notice no glue line, (the thin lines are veneers) and they have never parted in any way.
I use an applicator to spread the glue and clamp it in a jig which prevents it sliding apart. I buy it by the gallon, and when not in use leave the bottle inverted making sure the top is secure, ( does not thicken).
I like it.
John.