Yes it is true that Galadriel gave Gandalf new clothes, but he certainly didn't her help in creating a new staff, he was more powerful than her. Also, I don't believe that there was inate power in the staff but it was just used by Gandalf to channel his own power through. The fact that his staff emitted light in Moria (he didn't emit power through his staff after that) was because he used his own power to create the light, it had nothing to do woth the particular staff, he probably could have been able to do it with a twig if he wanted.
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His sword was long his lance was keen
His shining helm afar was seen
The countless stars of heavens field
Were mirrored in his silver shield
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