Ian Culverhouse once again fielded two different XIs - and his first half team trailed at the break courtesy of a Spalding trialist's effort on 20 minutes.
Woods whipped home an angled free-kick into the far top corner in the 64th minute, but Spalding won it seven minutes later through Sam Bennett's diving header.
United (first half): Cameron Gregory, Michael Bostwick, Jacob Hazel, Adam Marriott, Dan Mooney, Jordan Richards, Frankie Maguire, Zak Mills, Dylan Hill, Jacob Scott, Trialist A.
United (second half): Jake Lovelace, Cameron Green, Mitch Roberts, Tom Leak, Keaton Ward, Martin Woods, Tony Weston, Sam Osborne, Oliver Wilson, Ellis Perry-Burdett, Trialist B.
Referee: Dan England.
Attendance: 503.
PIC: Lee Keuneke.