Both constructions are correct, but not for the obvious reasons. Besides the missing article in these two sentences, the difficulty is with the verb itself.
- Since she started the journey, she has felt sick.
- Since she started the journey, she has been feeling sick.
The present perfect (PP): An activity that has been completed in the past.
- Its primary purpose is to emphasize that the action is completed.
- It is used to enumerate actions, to state achievements, state the unfinished.
The present perfect continuous (PPC): An activity that has just been completed and has an impact on the immediate moment or continues into the future.
- It has two primary uses. It can be used to emphasize the impact on the present, but it is also often used to emphasize the length of time for action. That latter is what is happening in sentence 2.
Option 2 leads us to empathize with the subject of this sentence:
Poor thing, she has been feeling sick since we left the house.
Whereas, the present perfect would do the opposite.
Fortunately, she has only felt sick once since we left the house. The journey has been quite easy for her.
So why is option 1 also correct?
Where everyone gets really confused is with respect to whether verb is active or stative.
The thing is that, when you are using a stative verb the PP is used instead of the PPC having the same meaning as the PPC. Stative verbs cannot be used with the continuous tense, regardless of whether it is present, future, past or perfect.
- I am happy.
- I was happy.
- I will be happy.
- I have been happy.
It would be incorrect to say:
- I am being happy.
- I was being happy.
- I have been being happy.
I think the point is clear.
The problem is with the verb to feel. In theory, it is a stative verb, but in practice it is used as both stative and active.
I feel happy.
I am feeling happy.
Understanding this, wouldn't the PPC be correct then? Not necessarily!
If you follow example 1:
- I feel happy. > I have felt happy.
The PP would actually be correct and PPC incorrect.
If you follow example 2:
- I am feeling happy. > I have been feeling happy.
The PPC would be correct, and the PP would be quite weird.
In conclusion, there's no avoiding it. Both are perfectly correct.