In English or Finnish I can say that I was touched by something or an experience was touching, meaning that I was touched emotionally, not physically. How can I express the same in Latin? Does tangere work this way, or should I use some other verb or perhaps other kind of word?
Certainly you may use tangere, though in this situation it and other verbs (moveo, commoveo, afficio, pelli and so on) usually have some sort of qualification:
amore puellae pelli ; amore fraterni commotus erat; etc.
This type of usage is actually quite common. And (of course!) it applies to emotions other than love:
pudore affixus est.
I have made up the examples. I hope that's sufficient explanation, without the need to quote actual instances?
I offer a quote from the Laudato Si encyclical, an official document of the Vatican (the country where Latin is, well, the official language). In English, it says:
With moving tenderness he [Jesus] would remind them [his disciples] that each one of them is important in God’s eyes
The Latin version says:
atque ipsis moventi blandimento memorabat quomodo suos ob oculos unaquaeque earum magnum habeat pondus
Thus, moventi, from the adjective movens, seems to be the word you are looking for. Wiktionary translate it exactly as "moving".
For something like this, I would use the verb moveō, or one of its derivatives. This can either take the emotion/result as its direct object (movet lacrimās "it moved [me] to tears"), or a metonymy like animum; when a person is the object, the verb is more often in the passive (movetur verbīs "he was moved by these words").
Moveō itself certainly works, but commoveō can be used for added intensity, and permoveō was seldom used for anything except this meaning (i.e. it almost never meant literal motion).