The way I handled it last time, for all I did anything with it, was the latter, full body swap: Crowley as Aziraphale in school, Aziraphale wherever Crowley is.

Which could be great if they need to transmit a message to the group in Hell, but then that puts the angel in Hell looking sus having to explain who he is and what he’s done with the real Crowley. The lucky thing is, the group would still have Alec so ...Alec could probably figure out what happened or verify him pretty fast.

In school would be easy. Aziraphale would explain how it backfired last time and whether that was his end goal this time or not, prepare Harry or someone with the info of what happened and why, for Crowley specifically, since this Crowley has never body swapped before.