If I exhaust Seraphine, Not Alone with Seraphine legend to play Raucous Anthem, can I opt to use Seraphine, Not Alone's effect first to play a Recruit token before resolving Raucous Anthem?

No — the Recruit token arrives after Raucous Anthem has already resolved, so it never gets readied or the +3 Might.

Exhausting Seraphine, Not Alone to help pay the Starry-Eyed Songstress cost does trigger her "when I become exhausted" ability, but abilities that trigger off a card being played only trigger once the act of playing that card is completed by the card's resolution (Rule 420.4.a). So Raucous Anthem resolves first, readying and buffing the units you already control at that moment; only afterward does Seraphine's trigger go on the chain and make the Recruit, which enters exhausted with its printed 1 Might. The +3 Might is a one-time "this turn" grant to the units present at resolution, so it doesn't follow units that enter later.

You can't reorder these — the trigger isn't placed on the chain during the cost-payment step, so there's no window to resolve it before the spell.