release-23.1: awsdms: use custom status checker

[blathers-crl]

Backport 1/1 commits from #108514 on behalf of @Jeremyyang920.

/cc @cockroachdb/release

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This commit switches the dms test to use a custom
status checker instead of the one from the SDK. The way the SDK checks for status changes has a likely chance of a race condition which causes it to return an error when there is really no error.

We now use a custom method to do the status checks which does an explicit status check each time the describe task api is called so we are always checking the latest result compared to the way the sdk was doing it.

Fixes: #108270
Release note: None

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Release justification: test fix

[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