Partridge gives this derivation chain:
sticky, adj. (Of persons) wooden, dull; awkward: 1881, Mrs Lynn Linton (O.E.D.). Ex stick, a dull person.—2. (Of stock) not easy to sell: Stock Exchange: 1901, *The Times, Oct. 24 (O.E.D.) : > by 1920, coll Cf. sticker, 1. q.v.—3. (Of persons) not easy to interview; unpleasant and/or obstinate; difficult to placate: from ca. 1919. Ex :—4. Of situation, incident, work, duty : unpleasant; very difficult : 1915 ('A sticky time in the trenches': O.E.D. Sup.); T. S. Eliot, in Time and Tide, Jan. 5, 1935, '[St Thomas of Canterbury] came to a sticky end.' This sense derives prob. ex senses 1 and 2 + S.E. (? orig.—1989—coll.) sticky, (applied to troops) apt to hesitate in obeying commands (O.E.D., adj., 2, 949 §2, b).