US Airways was the first.
"US Airways placed the first order for the re-engined (meaning from the 737-200 version) Boeing twin-jet by signing for 10 737-3B7's on March 5, 1981. The first 737-300 flew on February 24th, 1984 to what was known as US Air. It was then certified in November of the same year. US Air AND
Southwest had both placed the 737-300 into service in December of that year as well."
The 737-400 was launched by Peidmont Air, correct and US Air got it, when they took over. 735 was launched by SWA as well as the 737.