the readMe says:
All WFDEI data files include grid-box centre longitude and latitude (on a regular longitude-latitude 720 x 360 grid).
"a regular longitude-latitude .. grid" is likely a geographic lat/long grid; on what datum... who knows? try a sphere.
the "Generated grid information"
reinforces that impression: gridtype = lonlat
The 'gridsize = 6477' squares with the xsize = 127 and ysize = 51
xinc = 0.5 and yinc = 0.5, so the cells are half degree
The center of the lower left cell is xfirst = -125.37598, yfirst = 24.562113
(remember the "..data files include grid-box CENTRE longitude and latitude.." from the readme)
so, subtract 2.5 from the given x and y to find the lower left corner.
That is all pretty simple; the datum may be a problem
(self-identified scientists types tend to ignore such things)
give it a shot defining the datum as wgs84
and see how it lines up once you have it projected to your target.
If it is off, try defining the grid on a sphere (size is, again, anyone's guess)
if you do that, you may have to create a custom (dummy) transformation with all the values at 0
to get from the sphere to the NA83 spheroid.
if it is still off a consistant distance, you can build in a coordinate shift.
This may sound like it can be a little snakey (a lot snakey), but people that build these types of files often don't pay much attention to mapping conventions; they just want things to be consistant within their own user community....