I'd love to do a whole pile of teaching on evangelism from the Psalms. But right now I've only got time for a sentence.
Evangelism Psalms-style means joyfully declaring the true LORD of this world and His mighty deeds.
In other words it is praise. And conversion happens when they join in.
We can get so distracted by 'relevance' and 'reason' and 'human response'. But we actually scratch all these itches in a far more satisfactory way if we pass over them directly and devote ourselves to the central task. We say "Let me tell you the happy news of the true God of this world - this active, praiseworthy, saving LORD."
And it's not so much about explaining how to be saved but how He has saved.
They are saved when they find themselves caught up in this act of worship.