domain name UNC path is inaccessible if one domain controller goes down

There's not really enough information here to answer your question, but hopefully some of this can give you a better understanding of how Active Directory functions and maybe you'll be pointed on the right path.

For login scripts and the like, I use the path: \\mydomain.local\netlogon

mydomain.local is a DNS zone that lives on the DNS servers for your Active Directory. By default, new domain controllers register A records for themselves in this zone and they also register A records for the zone name itself as their IP. The records should be in the mydomain.local zone and look like this:

(Same as parent folder) Host(A) 10.10.10.2 <timestamp>
(Same as parent folder) Host(A) 10.10.10.3 <timestamp>

If you do nslookup mydomain.local the DNS server should return a list of all domain controllers. If not, you're missing an A record at the very least.

If one of those is missing, you should re-create it manually and do some root-cause analysis to find why it's missing. When a client tries to look up \\mydomain.local in this case, it is doing it by the A record.

You should also check the _msdcs zone and make sure that both servers are listed there where they should be. If you can log on while serverB is offline, then this zone is likely OK.

You should use dcdiag.exe just to make sure that both domain controllers are in fact healthy and functioning properly.

---

Also, if you're not running DNS on both DCs, you should and your clients should use both servers for DNS.

Also, the DCs should use each other as the primary server for name resolution and themselves as secondaries to prevent replication issues. Just to clarify, this means:

serverA
Primary DNS    10.10.10.3
Secondary DNS  127.0.0.1

serverB
Primary DNS    10.10.10.2
Secondary DNS  127.0.0.1

If you don't do this, you run the risk of breaking replication in certain situations.