Zhang and Du have been training in China since July, with athletics world governing body, the International Association of Athletics Federations (IAAF), requiring addresses for the two in order to perform random out-of-competition testing.
After warning the women several times, the SAA's patience finally wore through and the pair have been axed from the squad for the upcoming Asian Games in Doha, Qatar.
"We have been talking to each other over the phone since the news broke and we don't know what to do now," a tearful Zhang told Singapore's Today newspaper on Monday.
"We've no mood to do anything, we can't eat or sleep. We don't even know if we should continue training.
"Everything seems meaningless now. It's very painful and it feels like we're facing a firing squad," the 28-year-old added.
24-year-old Du has been training at the Beijing Sports University, while Zhang has been practicing at the Shenyang Sports School and both have appealed to the SAA for clemency.
"We did not deliberately do this as we didn't fully understand the requirements and implications," added Zhang, who is ranked fourth in Asia, one place ahead of Du.
"But it was our mistake and we hope the SAA will forgive us and let us compete at the Asian Games.
"We only want to represent Singapore and do well for the country."
The 15th Asian Games will take place in Doha from December 1 to 15.