In showdowns, do you just keep passing priority until both players have no more actions to play, or is it just one action per player?

Both players keep acting — a Showdown doesn't end after one action each. It only closes when all players pass Focus in sequence without playing anything (Rule 347.2).

After each chain resolves, Focus passes to the next player, who can play another Action or Pass. This loops until both players consecutively Pass on an empty chain — so you can trade Actions back and forth many times (FAQ #3157).

One catch: Actions can only be played on an empty chain. If something is on the Chain, only Reactions are legal. And "passing priority" during chain resolution is not the same as "Passing" Focus — that's the sequence that actually ends the Showdown (FAQ #218, #454).