Should We Believe the Claims of Fortunetellers and Soothsayers?

No, we should not believe them.

Evidence from the Noble Qur'an
{Say: None in the heavens and the earth knows the Ghayb (unseen)}, {Soorah an-Naml, verse 65]

Evidence from the authentic Sunnah
((Whoever goes to a fortuneteller or soothsayer, and believe what they say, has disbelieved in what has been revealed to Muhammad)), [Transmitted by Ahmad]