[3.11] gh-105375: Improve error handling in `zoneinfo` module (GH-105586) (#105613)

Fix bugs where exceptions could end up being overwritten
because of deferred error handling.

(cherry picked from commit 33c92c4f15539806c8aff8574ff30a8b307e3e4d)

Co-authored-by: Nikita Sobolev <mail@sobolevn.me>
Co-authored-by: Erlend E. Aasland <erlend.aasland@protonmail.com>