Here's the passage with grammatical corrections, preserving the original meaning, word choice, and style exactly as requested:
"I know I said I will be doing it today, but..." Claire's voice broke a bit as she hesitated, not sure how she was supposed to proceed.
She then gulped her saliva down before looking directly into my face.
"I'm sorry, I know this breaks the flow and immersion, but..." she bit down on her bottom lip with a look of unease on her face. "How do I begin?"
Looking up into the girl's face, I...
I actually wasn't sure how to react.
A part of me wanted to just laugh out. A different part of me heavily protested against doing so, fearful of having Claire feel bad about her attempt just because of a failed start. Then, there was a part in me that wanted to just hurry her up while yet another part was rushing to help her out.