Interesting that you mention this:
said by garrettm:
Third: yes. Still does not map, but comes back with something to the effect that this server does not take sharing (SMB)
Only Linux and Unix systems (including the Macintosh) use Samba (SMB) to map to Windows drives. Windows doesn't have a need or has a use for Samba. Are these *nix-hosted shares you're trying to map to on a Windows machine?
Also, Frontier stopped using Verizon IPs a while ago. You may however want to do a whois against your IP to see who owns your netblock. I'm pretty sure the sale of Verizon's network included some ranges in the form of SpinCo as a part of the Reverse Morris Trust deal. In which case, minus the routing differences between Frontier and Verizon along with the settings used, the IPs basically had a transfer and a change of records. The Actual IPs themselves never changed.
If AnyConnect is truly making a solid connection, whatever ISP you're on should not be affecting the connection. It's a tunnel after all. Unless it's an MTU issue or an ALG/MITM issue I'm failing to see how being on Frontier FiOS breaks things. Unless of course, the client is set up in a more unique stance where it uses the system DNS rather than the DNS supplied by the Gateway, or it isn't tunneling all network traffic through the VPN in the first place.