The OED Online, in an entry for 'moist' updated for the third edition, September 2002, addresses the issue directly, if somewhat hesitantly:
A. adj. 1. a. Slightly wet, imbued with moisture; containing liquid in a state of suspension or absorption; not dry; damp, humid.
In many contexts now differing from damp by having a neutral (sometimes, esp. with reference to the texture of food, good) rather than a negative sense, and in connoting a lesser degree of wetness. In early use the word had a wider application.
(Bold emphasis mine.)
The note concerning the greater degree of wetness connoted ("in many contexts") by 'moist' was added in the 2002 update. The note in the second edition (1989) was
Now differing from damp in having no tendency to imply either an undesirable or a merely temporary or casual condition. In early use the word had a wider application.
The three historical quotes from the 1900s given for the cited definition of 'moist' are from a US author (1901), an Australian author (1972), and the Toronto Star (Canadian, 1993). These quotes have no explicit bearing on the aforementioned observation concerning the "lesser degree of wetness" connoted by 'moist'.
