External Facing F5 DNS between data centers

We have external F5 DNS enabled at 2 of our datacenters. The public address spaces are being advertised out of both DCs, but we are looking into no longer doing that. I have 2 questions. First, is it reasonable to go after a public address self-ip on the F5 LTM guests. My concern is that the VIP itself would never actually be marked down if our internet circuit went down at a site. F5 DNS queries a private address on the LTM guests. If it went after a public address, then i would expect that F5 DNS would only be able to query a guest if it was actually reachable. If we stopped advertising public address space between DCs, then these queries would go out via the internet.

2nd question, can i build a health check for the LTM VIPs that would mark the VIP down if the local internet connection was down? Something like a query to the carrier's end of our connection with them.