Word for the sadness you feel when something so wholesome it makes you want to cry

I think your best bet is to say you found a behavior or story to be touching.

capable of arousing emotions of tenderness or compassion

This can be used of genuinely sad things as well, but it works with and is often used for being emotionally moved by good things as well.

Examples

(IJD)

One speaks of a story, of a letter, which are touching. “I am touched by your attention”.

(NIH)

Warm and touching tears: tearful individuals are perceived as warmer because we assume they feel moved and touched

(Quora)

Should I hold back my tears or just cry, for the touching or sad moments?

The first example established that it is used of positive things that produce a tender, sympathetic reaction. The second shows it's association with tears. The third distinguishes it from sadness.

I suggest the word poignant. Chambers gives the definition

touching, moving, exciting pathos; stinging, pricking; sharp; acutely painful; penetrating; pungent; piquant.

and you can have poignancy (n) and poignantly (adj) too. exciting pathos seems to be exactly what the questioner is asking for.

The OED gives a similar definition and offers examples of the word's use such as

Poignant memories of his father, whose painful boyhood..rendered him incapable of discussing his past.

Anne stood under the willows, tasting the poignant sweetness of life when some great dread has been removed from it.

and

There was in this composition such a fusion of sadness and dignity, so poignant an expression of the soul of oppressed and suffering humanity.