Await changing categories before changing query field.

Changing categories in the emoji picker will clear the query field,
so we have to wait until the category has changed before setting
the query field to the initial query.

Bug: b:350358944
Change-Id: Ieef8b56155af116426530eeb3bb9fdd1edfa4cf2
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5693361
Commit-Queue: Darren Shen <shend@chromium.org>
Reviewed-by: John Palmer <jopalmer@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1325938}