I don't think it is a short. No short is visible on the board and all the voltages measure as expected. The circuit works remember. It's just that the regulator is warmer than I expected.
I removed all chips from the board and also removed the 9V regulator. I took voltage measurements (in red below) on the resulting simplified circuit, and used those to calculate currents (in blue below). The 12V regulator was still warming up as before with this set up.

Voltages are reasonable and total current draw is 0.47mA + 0.49mA = 0.96mA.
The measured current into the regulator (using a multimeter) is about 10.5mA. It actually starts out at around 9.5mA and then slowly increases which I think is the result of the regulator heating up towards its final temperature.
The regulator datasheet says the regulator quiescent current is about 6mA but I don't know if the test circuit they used to measure that for the datasheet means I should expect the same sort of quiescent current in my example or larger.
Assuming quiescent current is 6mA, then I am still short of couple of mA. My scope and multimeter are not great quality so I don't know what the measurement errors are like.
The strange thing is that the regulator on my SAD1024 build doesn't get warm (even when I increase the clock rate to match the 3207 circuit). It's the same LFO/VCO remember. So maybe I have a bad regulator on my 3207 build. Unfortunately my soldering iron broke in the process of desoldering (handle actually snapped off and I was lucky not to burn my other hand) so my work on this is done.