When I am trying to say I'll hang on and persist through tough situations, I first thought of 'sticking through', but today when I think about it, I am not sure if this usage is correct so I found its definition https://idioms.thefreedictionary.com/stick+through, which isn't what I thought. Am I right about this usage?
You've confused stick through with the idiom you want, which is quite similar: stick it out.
to continue to the end of a difficult or unpleasant situation:
I know things are difficult right now, but if we just stick it out, I'm sure everything will be OK in the end.