Have you read
http://forum.tvfool.com/showthread.php?t=901
The grounding connector installed by Verizon sounds as if it has capacity to attach additional conductors without disturbing the existing connections. I would would be inclined to use that connector.
One ground connection to the coax shield should protect the entire system. A separate ground block for each distribution run is not necessary.
I recommend grounding the mast as one step and then grounding the coax shield as a second step. Each are independent ground conductors until they meet at the electrical service ground.