Radak does not say exactly what you said. Rather, he wrote (my own translation):
"Our Rabbis, za'l, deduced from this that a person (adam) is obligated to visit his teacher on Shabbat and the festival, as it states (in this verse) 'it is neither Shabbat nor Chodesh', implying that on Shabbat and Rosh Chodesh she would (be obligated to) travel to him (Elisha)."
The reference to Chazal saying this is to Rosh Hashana 16b and Sukkah 27b:
"ואמר רבי יצחק: חייב אדם להקביל פני רבו ברגל, שנאמר 'מדוע את הולכת אליו היום לא חודש ולא שבת' - מכלל דבחודש ושבת איבעי לה למיזל"
This does NOT say that women in general visited prophets in general, as a custom.
The Shunamite woman had a special connection to Elisha, in that he was reckoned to be her teacher / Rebbe. Thus he would always stay in her (and her husband's) home when he visited the area. (See II Kings 4:8.)
Based on Rabbi Yitzchak's statement, this would apply to any other person who had a similar connection to the prophet, as rebbe. But it is not cast as a custom for women, or for husbands accompanying their wives to visit the prophet.
Further, this is casting a rabbinic practice back to Biblical times, akin to other midrashim claiming that the patriarchs kept the entire Torah. If we choose to take this non-homiletically, then I do not think that we should recast it into some sort of proto-custom, involving women in general and prophets in general, and then seek to establish the contours of this reconstructed custom. There won't be any sources to establish those contours.
As to the reason for the obligation of visiting one's rebbe, see here at Virtual Bet Midrash, it is possibly an expression of kavod, honor. R. Yonasan Eybeschutz explained it as a replacement for visiting the Temple.