The additional information is helpful. You're correct that NADCON, specifically its CONUS (contiguous/continuous US) grid file. In ArcGIS, that's the NAD_1927_To_NAD_1983_NADCON transformation. Because NADCON is a piecewise method (in a way), bearings and distances won't be the same. To get those corrected, you would probably have to have a surveyor re-adjust/resurvey the data.
Melita