Hmm. Not quite. WADA sets the protocols, oversees the research, certifies the labs etc. etc. However, they do NOT do any testing themselves.
ASADA and equivalent organisations in each country do the dirty work in making sure the WADA code is followed and implemented by the national sporting bodies. For example, for the US version, USADA:
"Its work includes in-competition and out-of-competition testing, the results management and adjudication process, the provision of drug reference resources, the therapeutic-use exemption process, various scientific research initiatives, and athlete and outreach education."