Black To Play And Draw

BLACK to make his 31st move in Yaoyao Zhu (1842/165) - Spanton (1881/168), Northumbria Challengers round four.

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31...Qb3
Counterplay, as so often, seems to be the best try.
32.Qd6
Komodo10's choice. Stockfish10 narrowly prefers 32.Qd7, when the engines agree the mainline runs 32...Kg7 33.Kh2 Qxc3 34.Qe7 Qc2 35.Qxe5+ Kg8, with a slight edge for White, presumably thanks to better piece-coordination.
32...Re8 33.Qd7 Re6 34.Qd8 Kg7 35.Rd3 Rf6
The crisis for Black has passed, and by now I was hoping, without good cause, that I might even get an advantage.
36.f3 Qc2 37.Qd7
YZ offered a draw while I was thinking about my next move.
37...Qe2 38.Kh2 Qc2 39.Qd8 Qe2  ½–½