Thurmond actually resigned from the senate and ran again as a Republican in a special election... and his seat made no difference in the balance of power. He actually switched to the minority.
In 1953, Wayne Morris switched to the Democrats, but refused to make his switch effective until the following election because it would have flipped party control of the senate. The integrity of politicians has taken a real dive...
No, you're confused. When he first ran in 1954, he won as a write-in candidate (Democrat), defeating the regular Democrat. He resigned in 1956 (as per prior agreement) and won the Dem primary outright without a write-in, where he served without interruption until 2003. He switched to the GOP in 1964 but did not resign. However, SC Congressman Albert Watson, who was a DINO Goldwater supporter, resigned in early 1965 (after the Dems stripped him of seniority) and ran as a Republican, winning in the special a few months later where he remained until 1971 when he lost for Governor.