Attached is the doc from ASPRS / Clifford Mugnier details Ghana's geodetic control and history.
I see that your data appears to be old coverage export files. Perhaps you should import them into a fgdb first, before trying to get the underlying GCS / datum correct.
The doc (& the ArcGIS documentation) says that the classical datum in Ghana is Leigon (WKID 1159) and there is a geocentric transformation available in the system.
If you have the data already matching to within 60m, then you obviously have the correct projection system already. Just need to apply this transformation and see if the alignment improves.
Also in the doc, CM details a 7 param Bursa Wolfe transformation which he says greatly improves the accuracy. This is not in the ArcGIS system (at 10.3.1), so you would have to try this by creating a custom transformation.