Judge Dagne Melaku said the Supreme Court needed several weeks to look over the "bulky" case file to review the evidence against them, setting the next hearing for December 19.
Eskinder and Andualem were among 24 people convicted in June under Ethiopia's anti-terrorism legislation. Eskinder was jailed for 18 years, while Andualem was sentenced to life.
They appeared in court Thursday alongside two other defendants also appealing their guilty verdict.
Eskinder, who is representing himself, rejected the claim that he had links with the US-based Ginbot 7 group, considered a terrorist organisation under Ethiopian law.
"I was accused of being a member of Ginbot 7, but there was no evidence provided by the prosecution," he told the court.
Meanwhile, in a separate court, 29 Muslims charged last month with plotting "terrorist" acts appealed against the charges.
Defence lawyers said the charges were unconstitutional and used language not recognised under criminal law, including terms such as "jihad" and "extremism".
All 29 are charged with "intending to advance a political, religious or ideological cause" by force and the "planning, preparation, conspiracy, incitement and attempt of terrorist acts," according to court documents.
The accused -- including nine prominent Muslim leaders and the wife of a former minister -- were jailed following protests in July staged by Muslims who accuse the government of interfering in religious affairs.
Their hearing continues on November 30.
Rights groups have called Ethiopia's anti-terrorism legislation vague and accuse the government of using the law to stifle peaceful dissent.
All those charged under the law since it was introduced in 2009 have been found guilty, including two Swedish journalists who were sentenced to 11 years in prison, but later released by the government after serving 14 months.