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GH-33953: [Java] Pass custom headers on every request (#33967)



### Rationale for this change

Some flight requests don't send custom headers. This PR should fix that.

### What changes are included in this PR?

Ensure custom headers are sent across on every request.

### Are these changes tested?

No

### Are there any user-facing changes?

Custom headers should now be attached to every call.

* Closes: #33953
* Closes: #33839

Authored-by: Diego Fernandez <diego.fernandez@dbtlabs.com>
Signed-off-by: David Li <li.davidm96@gmail.com>