I couldn’t find any proof whether Dongwan was already a SM trainee back in 1996 but I’m pretty sure he wasn’t. Unlike current SM idols, there are no audition tapes of Shinhwa floating around to give us an idea of when they joined SM. Then again, Dongwan was cast off the street so that makes the date he joined even fuzzier. What I do remember is Dongwan saying he put aside his acting dreams to join Shinhwa, which makes me think he was already doing small acting jobs before he joined SM.
Also, just to clarify, he actually had roles in 5 different dramas/specials before he debuted with Shinhwa:
1996 - 성장 느낌 18 세 (SBS)
1996 - 스타트 (KBS)
1996 - 산 [Mountain] (MBC)
1996 - 마지막 승부 (School Special)
1997 - 내안의 천사 (KBS)
Credit: Source1, Source2, Source3