It sounds like you have split-horizons DNS, where your AD namespace and public-facing namespace are the same. If this is the case, you'll need to make an A record for
www in your internal AD zone and set the correct external IP. Without this, internal clients won't be able to resolve it. This is because you have two sets of servers with disjoint information that are both authoritative for your DNS zone.
This is the exact reason that Microsoft recommends using an unused third-level domain name for your AD namespace. For example, if you own
mycompany.com and you host public websites on it, you should use something like
internal.mycompany.com for your Active Directory namespace. If you can do a domain rename (i.e. you have a simple client/server infrastructure with no Exchange in the domain), you should look into it. If you can't do a domain rename to fix this, you're in for some headaches.
Edit: Per your update, you should absolutely redo this since the install is only 24 hours old. Absolutely no doubt about it. Having a split-horizon DNS situation is never a good idea when it can be avoided. You can still have
mycompany.com be the NetBIOS name of the domain, so users see
MyComapny\User instead of something like
Corp\User, but the FQDN of your AD namespace and the FQDN of your public facing DNS namespace should not be the same.