[Frontend] Adjust try/except blocks in API impl

[njhill]

These were changed to separate blocks in #9759 but I feel it's cleaner/clearer as a single block. It actually doesn't matter which parts of the block raise the specific exceptions in the except clauses, we still want to handle them in the same way.

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[DarkLight1337]

I don't mind either way, but maybe we should also do this for chat endpoint to be consistent? (Originally I did it the other way round)

[njhill]

Thanks @DarkLight1337 ... I think the chat endpoint is already like this (at least there aren't equivalent separate try/except blocks in the non-streaming impl).