Wa alaykum salam wa rahmatuLlahi wa barakatuHu,
Public displays of intimacy between a husband and his wife should be avoided. In Tuhfah 10/225, Ibn Hajar mentioned that one kissing his wife on the mouth is impermissible, but not on her head. And touching her chest would be impermissible. Similarly to touching her chest, touching other parts that are places on a women's body one enjoys sexually is unlawful.
I am not aware of a clear naql in the Shafi'i School's fiqh books on holding one's wife's hand. The discussion on this topic comes in the Book of Testimony, under what violates one's maru'ah. In Mughni al-Muhtaj 6/339, Khatib Shirbini defines maru'ah as, 'An individual acting according to the standard of his peers, those who uphold the way of the Sacred Law and its etiquettes, in the place and time he is in.' Khatib continues to say that matters based on custom are hard to determine as they change according to conditions, times, and places. At present, in the United States, those who would be considered in the formation of custom do not hold hands with their wives in public. Thus, it is prudent to avoid it.
And Allah knows best.
Answered by Shaykh Yaqub Abdurrahman