Oath of office (Senate)
I do solemnly swear (or affirm) that I will support and defend the Constitution of the United States against all enemies, foreign and domestic; that I will bear true faith and allegiance to the same; that I take this obligation freely, without any mental reservation or purpose of evasion; and that I will well and faithfully discharge the duties of the office on which I am about to enter: So help me God.
Should he resign?
he is not constitutionally prohibited from holding his senate seat. He sure as heck is subverting the constitution by running for President though.