Here we go... Turns out Google Ngram searches are case-sensitive, so I capitalized "Try" and, err, tried again.
Here's a use of "Try to keep up!" as a standalone phrase in 1864, although it's apparently being used in the sense of keeping up one's composure rather than keeping pace with someone. And it's the only instance I could find. But it and the other phrase I cited do demonstrate that "keep up" did exist at the time as an intransitive phrase. So I don't see any reason why Dottie couldn't have used the phrase in 1947.
In general, a lot of things that people think are recent tend to be a lot older in reality. So it's generally best to be skeptical of the assumption that something is recent. Like how people tend to assume it's a recent trend to use "literally" to mean "figuratively," when in fact it goes back centuries.