"I have no idea if it's actually Alan singing" teleadm
I found this:
Ladd does indeed sing it, as confirmed in Gene Feldman's documentary "Alan Ladd: The True Quiet Man" (1999), now available to view on Amazon Prime.
Just about all stars back then were trained to be tripe threat performers as in act, sing and dance so its not surprising that Ladd could sing and dance. Hollywood during this time actually had people with talent.
a wise man once said "If someone doesn't want responses to his/her posts, then s/he shouldn't post them. If someone wants specific users to not respond...well, that's too bad." (link)