How did john adams persuade the delegates to sign the declaration of independence?

He was THE major mover-and-shaker in the Second Continental Congress from 1775-6 busy convincing delegates to vote FOR independence. 

But he also knew that he was somewhat abrasive and that some SOUTHERN delegates were not that interested in following the lead of someone from Massachusetts (folks who were widely considered the 'hotheads' in rebellion and pushing for independence). So it was he who encouraged VIRGINIANS to take the lead -- first by getting Richard Henry Lee to obtain from the Virginia legislature a RESOLUTION asking Congress to vote for independence, which Congress did vote for on July 2. 

Now Lee would probably have been automatically made head of the committee to write a document EXPLAINING the reasons for the vote for independence (the formal "Declaration of Independence"), but family health issues kept him away and he was replaced by another Virginian, Thomas Jefferson, who (as committee chairmen usually did) drafted the document.