Can't remember who sang it, but it was popular in summer of 1952. I was in ROTC summer camp in Valdosta, GA. Hot as the hinges of Hell. We'd come in from marching or whatever, head straight for the shower, strip off our fatigues, and shower, while singing that song. Crazy? Probably, but I still can't take a shower without thinking of that song.
Joe, I'm new on this thread but thought I'd jump in and see if one of these two recordings I found on YouTube were what you remember.
Teresa Brewer was the first artist to record "Gonna Get Along Without You Now" in 1952.
Then Patience and Prudence enjoyed even more success with the song when they recorded it in 1956. In 1957 it hit the #11 position on the charts.