If you visit a “soothsayer”, what you’re looking for is a prediction of the future. You might or might not find the prediction soothing — a word that comes from the same origins.
“Sooth” is an ancient word meaning truth. It was originally Old English, and by the 1700s it was already archaic and obsolete. Then in the early 1800s Sir Walter Scott resurrected it to use in his historical fiction, and ever since it’s been popping up here and there, either to give a story an authentic patina or, as PG Wodehouse used it in the 1930s, for humor: “‘Is one to have no privacy, Glossop?’ I said coldly. ‘I instructed Jeeves to lock the door because I was about to disrobe.’ ‘A likely story!’ said Tuppy, and I’m not sure he didn’t add ‘Forsooth!’”
“Soothsayer” appeared in the 1300s, when it simply meant somebody who told the truth. It wasn’t until a century later that it started to be used to mean fortune tellers and the like; people who told the truth “about the future.” If some soothsayer saw that coming, though, they didn’t write it down.
“Soothe” has a very different history, even though it began as the exact same word as “sooth.” In Old English, “soothe” was just a different form of “sooth,” and meant verifying the truth. Sometime around the 1500s, “soothe” started to mean to attest to something, to agree that what somebody said was true: “Then must I sooth it, what euer it is: For what he sayth or doth can not be amisse” (1588).
Then in the 1600s, if you “soothed” someone, you were humoring them, even if you knew what they said wasn’t true: “If a private friend admonish not,..but if he sooth him, and allow him in his faults” (1644). This sense of “soothe” persisted for century or so, then came to mean whatever you had to do to calm somebody down. That’s how Jane Austen used it in 1814: “Was he only trying too [sic] soothe and pacify her, and make her overlook the previous affront?”
That’s how “soothe” is used today, and it doesn’t just apply to people; you can soothe an animal, and there are any number of ads pointing out that individual parts of your body can be soothed with products ranging from throat drops to hot tubs. There’s no record of any soothsayers predicting that, either.
So “sooth,” which meant truth, disappeared for centuries only to be revived as an antique curiosity and source of humor, while “soothe” has been used right along, but went from meaning truth to meaning making somebody feel better. Maybe this is because everybody finally realized the truth hurts. In any event it ain’t what it used to be, forsooth.