[verifier] migrate off deprecated ProcessAdapter

[pq]

Tidy up some lingering ProcessAdapter references.

(Wildcard imports borked my previous grep -- note to self: don't use grep AND avoid wildcards!)

See: #8764

---

• I’ve reviewed the contributor guide and applied the relevant portions to this PR.

Contribution guidelines:

• See our [contributor guide](https://github.com/flutter/flutter/blob/main/docs/contributing/Tree-hygiene.md for general expectations for PRs.
• Larger or significant changes should be discussed in an issue before creating a PR.
• Dart contributions to our repos should follow the Dart style guide and use dart format.
• Java and Kotlin contributions should strive to follow Java and Kotlin best practices (discussion).