Add missing locks when update settings table

[rumyantseva]

Description

Closes #1206.

Note: I didn't refactor how the settings table is used (as it's not in the scope of the task), I only fixed the exact problem with missing locks.

Readiness checklist

• I added tests for new functionality or bugfixes.
• I ran task all, and it passed.
• I added/updated comments for both exported and unexported top-level declarations (functions, types, etc).
• I checked comments rendering with task godocs.
• I ensured that the title is good enough for the changelog.
• (for maintainers only) I set Reviewers (@FerretDB/core), Assignee, Labels, Project and project's Sprint fields.
• I marked all done items in this checklist.

[vercel]

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Name	Status	Preview	Updated
ferret-db	✅ Ready (Inspect)	Visit Preview	Nov 14, 2022 at 10:27AM (UTC)

[codecov]

Codecov Report

Merging #1381 (7303e8f) into main (b4e281e) will increase coverage by 0.10%.
The diff coverage is 66.17%.

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #1381      +/-   ##
==========================================
+ Coverage   67.26%   67.37%   +0.10%     
==========================================
  Files         279      279              
  Lines       13325    13353      +28     
==========================================
+ Hits         8963     8996      +33     
+ Misses       3432     3425       -7     
- Partials      930      932       +2     

Impacted Files	Coverage Δ
internal/handlers/pg/pgdb/collections.go	50.00% <50.00%> (ø)
internal/handlers/pg/msg_listdatabases.go	66.99% <56.25%> (-0.41%)	⬇️
internal/handlers/tigris/tigrisdb/errors.go	53.57% <60.00%> (+1.39%)	⬆️
internal/handlers/pg/pgdb/settings.go	44.21% <63.63%> (-1.65%)	⬇️
integration/helpers.go	87.62% <100.00%> (+0.39%)	⬆️
internal/handlers/tigris/msg_create.go	74.64% <100.00%> (+8.97%)	⬆️
internal/util/version/version.go	48.52% <0.00%> (-4.42%)	⬇️
internal/handlers/common/filter.go	84.65% <0.00%> (ø)
... and 4 more

Flag	Coverage Δ
integration	63.49% <66.17%> (+0.11%)	⬆️
mongodb	14.07% <7.35%> (-0.04%)	⬇️
pg	52.58% <54.41%> (+0.02%)	⬆️
tigris	35.34% <14.70%> (+0.03%)	⬆️
unit	26.06% <37.09%> (-0.04%)	⬇️

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