Both fell to Chinese opponents in the afternoon session on Friday, with Drago whitewashed 4-0 by Liang Wenbo while multiple-time World Championship runner-up White was ousted 4-2 by Zhang Anda.
Fellow veteran Steve Davis progressed with a 4-2 triumph over Kurt Maflin in the same session, and Nigel Bond had eased through 4-1 against Joel Walker in the morning.
Marco Fu, losing finalist in last week’s German Masters to Ali Carter, saw off Passakorn Suwannawat 4-1 to advance in the evening session which also saw Mark King and Mark Allen move on to the next preliminary round.
The Welsh Open runs from 11-17 February, with a total prize fund of £250,000.
Rod Lawler 4-3 Li Yan
Nigel Bond 4-1 Joel Walker
Peter Lines 1-4 Pankaj Advani
Thepchaiya Un-Nooh 4-3 Michael Leslie
Mark Joyce 4-2 Tian Pengfei
Anthony Hamilton 4-1 Sean O'Sullivan
Liang Wenbo 4-0 Tony Drago
Alfie Burden 4-3 Scott Donaldson
Jimmy White 2-4 Zhang Anda
Alan McManus 4-2 Jamie O'Neill
Jimmy Robertson 4-3 Robbie Williams
Steve Davis 4-2 Kurt Maflin
Mark King 4-0 Aditya Mehta
Marco Fu 4-1 Passakorn Suwannawat
Martin Gould 3-4 Sam Baird
Robert Milkins 4-2 Paul S Davison
Mark Allen 4-2 Barry Pinches
Jamie Cope 0-4 Ian Burns