Liz,
I've been digging around a bit more on this and I think that the Harington we are looking for could be of a much earlier generation. The Elizabethan Haringtons you refer to, Sir James and his son Sir John, later Baron Harington, used "sable a fess argent" which seems to have been in their family from at least 1308 when it is recorded at the Dunstable tournament.
There are however, other Haringtons from the Rutland and Leicestershire area who used variations of the "Gules a chief or, a bend azure " with differences, which appear in the Visitations of the early 1600s. This suggests an earlier joint ancestor and a clue may be the church at Glaston where a Hugo de Harington was installed as patron c1220 and the arms "Or, a chief gules, on a bend azure an annulet or" appear in the church, though not directly connected to Hugo.
In the church at Staunton(Stonton) Wyville is the monument of Wm Wyville (d1452) showing his arms impaled with (undifferenced) Harington. So his wife was a Harington of the line we are looking for.
I am wondering if a Harington had something to do with the rebuilding in the mid-1400s ??
Also : you said you had a mid 18th century description of the arms. Can you please confirm that the blazon of the three charges on the chequy coat describes them as "quatrefoils". ?
Regards
Maec