release-22.1: ccl/sqlproxyccl: fix TestConnectionMigration test flake

[blathers-crl]

Backport 1/1 commits from #83457 on behalf of @jaylim-crl.

/cc @cockroachdb/release

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Fixes #83096.

It appears that database/sql does not provide any thread-safe guarantees for
methods on the Conn type except the Close method. This commit fixes a test-only
issue that causes a panic when there's a race between internal Conn methods
by ensuring that Conn methods are used in a single-threaded way.

Release note: None

Release justification: sqlproxy only test change.

[blathers-crl]

Thanks for opening a backport.

Please check the backport criteria before merging:

• Patches should only be created for serious issues or test-only changes.
• Patches should not break backwards-compatibility.
• Patches should change as little code as possible.
• Patches should not change on-disk formats or node communication protocols.
• Patches should not add new functionality.
• Patches must not add, edit, or otherwise modify cluster versions; or add version gates.

If some of the basic criteria cannot be satisfied, ensure that the exceptional criteria are satisfied within.

• There is a high priority need for the functionality that cannot wait until the next release and is difficult to address in another way.
• The new functionality is additive-only and only runs for clusters which have specifically “opted in” to it (e.g. by a cluster setting).
• New code is protected by a conditional check that is trivial to verify and ensures that it only runs for opt-in clusters.
• The PM and TL on the team that owns the changed code have signed off that the change obeys the above rules.

Add a brief release justification to the body of your PR to justify this backport.

Some other things to consider:

• What did we do to ensure that a user that doesn’t know & care about this backport, has no idea that it happened?
• Will this work in a cluster of mixed patch versions? Did we test that?
• If a user upgrades a patch version, uses this feature, and then downgrades, what happens?

[cockroach-teamcity]

This change is