I'm wondering if we don't have a mixture of share-level permissions and NTFS permissions at the server. When you say that the global group is on the share, do you mean that you had the share interface open and you clicked on permissions? If so, the underlying NTFS permissions on the folder might be blocking you.
The standard way to handle share permissions is to leave the everyone group with full access to the share then modify the NTFS permissions to control access. Put the global group on the NTFS permissions for the underlying folder and see if that solves the problem.
If I totally missed the mark, mail me back with screen shots of the share permissions, the NTFS permissions and a listing from the net group command. Also, make sure that the server has no event log entries indicating that it has trouble communicating to the domain. This includes any WINS errors.
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