Building a Lightsaber with pretty much on electrical experience
First off, 3 AA batteries &amp; 3 C batteries both (when wired in series) will produce ~4.5V (~3.6V if you use NiMH chemistries), but 6 AAA batteries (series wired) will produce ~9V (~7.2V NiMH). Using double the voltage will often cause serious damage to your device(s), and potentially to your person as well!

Please disconnect the 6 series-wired batteries before you have a fire start inside the toy.

Now, for why the batteries get soo hot; AA batteries are capable of producing much more current (I'll guess around 3x as much) than AAA batteries without heating up (due to lower internal resistance), and C batteries much more again than AAs (for the sake of argument, we'll assume 3x as much again here). Now, when you combine a device with C battery current demands in parallel with one which demands AA battery current, you could be demanding as much as 12x more current than those poor little AAA batteries were designed to efficiently put out! (and that's ignoring the excess current being forced through the circuitry by doubling the voltage)

Your "best bet" for running these 2 circuits would be to use the 3-series C batteries from the larger device, or mabe upgrade to a 3-series D battery pack if the C batteries don't last as long as you would like.