"She feels alright."
Firstly, I want to discuss about the English sentence. Does "She feels alright." also mean "She feels that it is alright."? I am not native English speaker so I can't tell, but I doubt it.
觉得 and 感觉 can often be used interchangeably, but there is a subtle difference. Comparatively, 觉得 is more objective, assertive and determined, and 感觉 is more subjective and uncertain.
When we make a judgement we often would use 觉得, e.g. 我觉得他的书毫无价值/I think his book is worthless. 感觉 is not very suitable here, for the same reason that we tend not to say I feel his book is worthless. Such statement would appear very harsh and direct. In such case using 感觉 instead would make us more like an observer, e.g. 我感觉他的书没有什么价值/I feel that his book doesn't have much value.
(Why people keep giving down votes to questions wo a reason? Nobody can ask a question?)
I also would have used 感觉 here. I also did not know that 觉得 could be used without reference to a subject matter.
But, about the difference: 觉得 can mean "think" or "feel." 感觉 is more squarely "feel." I don't think it can mean "think" (ei, it's emotions and perceptions, not logic).