Hamilton grabbed a stoppage time equaliser to deny Livi all three points at New Douglas Park.
Ali Crawford and May had early chances that drew good saves from Andy McNeil - the first coming after just 17 seconds.
But Livi took control of the first half thereafter, with efforts from Stefan Scougall, Jesus Garcia Tena and Liam Fox all failing to find the net.
David MacKinnon's low shot early in the second half was well tipped wide by mcNeil, while Scougall was crowded out in the Accies box after bursting free of their back line on 68 minutes.
The visitors had enjoyed plenty possession without testing home stopper Kevin Cuthbert too often, but they grabbed the opening goal with twelve minutes left on the clock.
Good work down the right from Danny Mullen eventually got the ball to Russell six yards out, and he defied the efforts of Cuthbert to force the ball over the line.
If that was a scrappy goal, the home side's dramatic equaliser was of a similar mould.
Play was well into time added on when the ball spun loose 10 yards out from the visitors' goal, and May's scuffed shot spun off a Livi boot and McNeil's fingertips before trundling over the line.
Cuthbert, Gordon (Fraser 88), Hendrie, Canning, Page, Routledge, Crawford, Fisher (Longridge 75), May, MacKinnon, Gillespie
Subs not used: Devlin, Docherty, Christie
McNeil, McCann, Booth, Garcia Tena, Barr, O'Brien, Andreu, Fox (Mullen 75), Ky.Jacobs (McNulty 64), Scougall, I.Russell
Subs not used: Morton, Talbot, Jamieson
Referee: Matt Northcroft
Official Attendance: 914
Match report: Calum Leslie