In this answer I am going to discuss the particular case of the word 躊躇, although it may extrapolate to other similar terms.
Examining example sentences
If you look up example sentences using 躊躇, you will see that the word can used in either of the two ways you describe:
- The action of hesitating or state of being hesitant
- Hesitating about something in particular
I picked some example sentences from the provided link for both usages:
Sense 1
ハインズ氏は少しぐずぐずと躊躇していた。Mr. Hynes hesitated a little longer.
もし何か質問があれば、躊躇せずに私に聞いてください。If you have any questions please ask without hesitation.
Sense 2
私はそれに躊躇していました。I hesitated about that.
助けを求めるのに躊躇しないで下さい。Don't hesitate to ask for help.
このことを彼女に伝えようかどうか躊躇している。I am hesitant about whether or not I should tell this to her.
Ambiguous
これを見て、わたしは躊躇しました。
If we examine the examples for Sense 2, the thing that one is hesitating about is marked by the particle に, although other particles such as かどうか also work here. I also chose a more ambiguous sentence, to highlight that context is crucial. In that last case, we can't really tell if わたし is hesitating about something in particular or not because we lack context. For example:
[Context: I'm about to cross a street when the traffic light is blinking and about to turn red, Sense 1]
これ[信号]を見て、わたしは[道を渡ろうか]躊躇しました。
[Context: I'm about to give a presentation in front of an audience and it turns out there is way more people than I expected, which makes me hesitant, Sense 2]
これ[観客]を見て、わたしは躊躇しました。
Please note that some sources for the provided example sentences are not reliable, particularly the ones coming from Tanaka corpus, because they are not necessarily written by native speakers nor being subject to curation or revision.
Examining your particular sentence
未だ見ぬ金閣に愈接する時が近付くに連れヽ私の心には躊躇が生じた。As the time to finally touch the Golden Pavilion, which I hadn't seen yet, got closer and closer, hesitation arised within my heart.
I'm inclined to believe that in this sentence, 躊躇 is used in the sense 1 specially because the emotional part of it is emphasized by saying that the feeling arised in one's heart (心には躊躇が生じた). Also note that 躊躇 is used as a noun here, not as a verb. To use it in sense 2. it would probably require to use it with する:
未だ見ぬ金閣に愈接する時が近付くに連れ、実際に行くのに躊躇し始めた。As the time to finally touch the Golden Pavilion, which I hadn't seen yet, got closer and closer, I started to hesitate if I should go.
Takeaway
The word 躊躇 can be used in both ways, either to convey a feeling of "hesitation" or to signify "hesitation to do something or about something". In some cases it would be clear (for example if the particle に is used), but there might be other instances where context is absolutely necessary to determine if its one usage or the other. As in many other aspects of Japanese, context is essential.