Reading back over the section it says nothing about Gandalf dropping Glamdring, so he must still have it for battling the Balrog.
Anyhow, I would say that the breaking of a staff is a result of the "breaking of power", if you will, but the staff is not merely symbollic.
Quote:
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"The staff in the hand of a wizard may be more than a prop for age," said Hama.
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Then again before King Theoden:
Quote:
He raised his staff. There was a roll of thunder. The sunlight was blotted out from the eastern windows; the whole hall became suddenly dark as night. The fire faded to sullen embers. Only Gandalf coud be seen, standing white and tall before the blackened hearth.
In the gloom they heard the hiss of Wormtongue's voice: "Did I not counsel you, lord, to forbid his staff?..."
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I think that these to quotes from TTT, Book 3, Chapter 6, would show that the staff is more than symbolic. I would say that it is like a channel for their power, and while they are not completely powerless without their staffs, it is certainly a part of their power.