Why are the NAD 83 position values so far from the NAD 27 values? Were the old coordinates wrong?

The old coordinates were not wrong, just different.

Positions obtained using the North American Datums of 1927 (NAD 27) and 1983 (NAD 83) are based on different earth shapes--or ellipsoids--and used the best technology available at the time. Mathematically, NAD 83 is a stronger datum because all previously existing horizontal stations and newer GPS surveyed stations were adjusted simultaneously. The positions within NAD 27 were adjusted in arcs, as the networks progressed across the country. Errors between stations adjusted in different arcs could have been substantial.

This issue is of declining importance and is seldom relevant to anything other than historical USGS maps (generally meaning maps published before 1990). All modern maps and GIS data are cast on NAD 83 or WGS 84, which are equivalent datums at map scales of 1:5,000 and smaller.

All federal agencies will replace NAD 83 and NAVD 88 (a vertical datum) with a new datum in the near future.

Learn more: Datum Shifts and Digital Map Coordinate Displays