I have not found any evidence of Elsie Carlisle’s having recorded “I Want Somebody to Cuddle Me”; she must have sung it on the radio. There is a very nice version of the song (going under an alternate title) by Nat Shilkret and His Orchestra, with Belle Mann on the vocals:
“I Want a Daddy to Cuddle Me” -- Nat Shilkret and His Orchestra
Transfer by Clive Hooley (YouTube)