In searching for the original "Sweet Rosy (or Rosey) Morning", I've only been able to come up with obsolete links and index entries. These have pointed to a song "Damon and Phoebe". (When searching, note that the o and e may be ligated as the special character œ, and "and" may be replaced by an ampersand.) The first lines run:

When the sweet rosey morning first peep'd from the skies
A loud singing lark bade the villagers rise;

www.colonialdancing.org provides an index entry for this song, though the "incipit" indicates a different tune entirely.

I also found this at the University of Buffalo's British Music Collection index:
721. PRING, Jacob Cubitt, 1771-1799.
Damon and Phoebe, a favorite Song with an Accompanyment for the Forte Piano...
London: J. Bland [1788].
3 pp.
CPM Vol. 46, p. 110. BUCp. 810. RISM P5461 (only one copy recorded).

Can someone provide the Broadwood version of this song, or the full text of "Damon and Phoebe"?