[release-22.0] CI: Address our two flakiest tests (#19587)

[vitess-bot]

Description

This is a backport of #19587

[vitess-bot]

Hello @mattlord, there are conflicts in this backport.

Please address them in order to merge this Pull Request. You can execute the snippet below to reset your branch and resolve the conflict manually.

Make sure you replace origin by the name of the vitessio/vitess remote

git fetch --all
gh pr checkout 19594
git reset --hard origin/release-22.0
git cherry-pick -m 1 644e9665605d463df25ef4322fc842853b5fb8de

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 69.47%. Comparing base (992254b) to head (8393c58).
⚠️ Report is 5 commits behind head on release-22.0.

Additional details and impacted files

@@              Coverage Diff              @@
##           release-22.0   #19594   +/-   ##
=============================================
  Coverage         69.47%   69.47%           
=============================================
  Files              1601     1601           
  Lines            213700   213704    +4     
=============================================
+ Hits             148470   148481   +11     
+ Misses            65230    65223    -7     

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