I watched that debate also. She has an oddly emotional speaking style; it often sounds like she’s on the verge of tears. I don’t know about “fussy”, but “easily rattled” or just “rattled” seems to fit. Everything she says sounds like she’s delivering a eulogy for someone that she deeply cared about. I guess the description I would use is “sad-sounding.” She speaks as if she is sad.
I know a lot of people might think that I’m being sexist against her for sounding “emotional” but, believe me, if a man spoke in this way, I would not rate his chances of success against Trump highly either. I’m also not against sounding emotional, in the right way. But this is the wrong way. Passionate is one thing, but sad-sounding is another.