Overton cleaned up captain Tim Paine and the Cricket Australia XI were an embarrassing seven down for only 25 when James Anderson had Milenko caught behind, but Matthew Short (28 not out) and Sandhu (17) ensured there would be a fourth day after England took the extra half-hour.
The veteran opener's dismissal triggered England's slide - he was followed to the pavilion by Mark Stoneman, who made 51, his third consecutive half-century on tour.
Jake Ball is in a race against time to be fit for England's opening Ashes Test with Australia after being ruled out with a troublesome injury.
From 0-79, the tourists lost seven wickets in a span of 17 overs, including losing four wickets for three runs.
However, the likes of James Vince (33) and Chris Woakes (33) made starts before losing their wickets, while Alastair Cook only managed 15 as England closed on 278 for 8 on the first of four days in Adelaide.
NSW seamers Simon Milenko and Gurinder Sandhu ripped through the Englishmen - Milenko has 4-32 and Sandhu 2-47.
Joe Root's side are already without Middlesex bowler Steven Finn, who was forced to fly home earlier this week after sustaining damage to cartilage in his left knee.
Suspended all-rounder Ben Stokes is unable to fly out to Australia while the investigation surrounding his recent arrest in Bristol remains ongoing.
Only a fightback from Jonny Bairstow (61no from 66 balls) lifted England to 207 all out, with injured paceman Jake Ball not taking the field as he awaited scan results on his sprained right ankle.
Moeen Ali, meanwhile, is yet to feature in the warm-up matches as he continues to struggle with a side strain.