From where I'm standing, the servo and motors are running just fine with a 9V battery power supply.
If they are working fine why are you posting here that you have a problem?
It's only doing so when a DC motor is running though, and the servo and motor have a different power supply, so this can't be it.
The DC motor generates interferance and that is what is causing the twitching. Do you want to cure it or not?
Or am I wrong in assuming that the RFI can't carry over several cm?
Yes you are wrong. How far away is your local TV transmitter, more than a few cm I would guess.
Isn't that a little overkill for 3 tiny motors?
No.
I don't have any capacitors, so I can't try this
Then either get some, or take up knitting, maybe electronics is not for you.