The Doe or Die motto refers back to the family's origins with the Douglas of Cavers family.

A revised entry for that family is being prepared at the moment.

I do have a variety of armorials for Springwood scattered through the Douglas Archives, so perhaps a task is to consolidate them. E.g. here:
https://www.douglashistory.co.uk/history/georgedouglas5.htm 

The Earls of Douglas—with one exception—inscribed no motto on
their seals. The exception is found on the seal (1373) of Archibald,
Lord of Galloway, who became third Earl of Douglas in 1388, where
from the beak of a peacock's head, which forms the crest, proceeds a
scroll inscribed with " some words that have not yet been deciphered."
l
Save in regard to the first, sixth, and seventh characters, the letters
of the legend are distinct. From a comparison of the difficult first
character, which resembles a Greek minuscule Omega, with the letters
beginning the word " Douglas " in the legends on seals of the 4th Earl of
Douglas and of Princess Margaret of Touraine,2
I am inclined to view it
as a monogram representing the letters D, 0,—the up-stroke of the small
D having been destroyed, or originally merged in the border of the scroll.
The motto then reads :—" Do batt dye." The strange word' " batt " seems
to be the old " Bout, But, Bond, Bit, Bud, Boost, v. imp.,—Behoved,
Was under the necessity of. ' He boot to do ' such a thing, he could not
avoid it. ' It bit to be,' it was necessary that this should take place "
(Jamieson, Scottish Dictionary). Thus the motto may be interpreted—
Do (daring deeds) though to die be the inevitable end. In archaic wording, we have here the chivalrous motto of the Douglases of Cavers themselves—" Do or die," and we are strengthened in the opinion that if the
Black Douglases had a motto at all, it was none other than this.3

The origins of Douglas mottos might make an interesting article.

Anyone up for it?