"Waxes cold," as waxes cauld, is in a version of "Waly, Waly" published in 1733.
See Child 204 in Bronson, p. 362, 2nd verse:

Waly, waly, gin love is bonny,
A little while when it is new;
But when it's auld, it waxes cauld,
And wears away, like Morning Dew.

Wax means to grow, or to become, or to increase. Weaxan is the Old English, the word is hoary with age and saddled with much use.

I also like verse 4:
When cockle-shells turn siller Bells,
And muscles grow on ev'ry Tree;
When Frost and Snaw shall warm us a',
Then shall my love prove true to me.

These verses read well, as well as sing well, and through exaggeration leave the reader with a smile.
When metaphors are stretched too far, they tend to lose effect on second thought, and the intent is lost.

The line about dew fading is in a 1725 version, I think. The verses above are close to those in Waly, Waly 2 in the DT.

Gee, folks, my eyes started to glaze over at the higher math and profundities in your posts. Something simple is the best way for A to reach C.