Cliff “Ukulele Ike” Edwards became the voice of Jiminy Cricket for Disney.
Phil Harris was another singer who became a voice actor for Disney (notably Baloo the bear in The Jungle Book).
True. I don’t think he’s all that bad on Hannah Montana, either - he plays a (stereo)typical sitcom dad, but does it well, and comes across as warm and sweet without being emasculated. Even makes fun of his “Achy Breaky”-period persona in one episode.
Shirley Temple, or did she start doing both at the same time?
Temple started as an actress who sang and danced in her films. Her first films were shorts shot in 1932, when she was 4 years old. I don’t think she had any time to be a non-acting singer first. 
He also played Elvis in Walk Hard, and a couple other minor roles.