Connect an electrolytic capacitor of some value to the manual reset of the 707. Don't forget a pull-up resistor if this is not already present. Let the RC time extend the (manual) reset way past the rise-time of the power-supply. Start with 10K/10uF and enlarge cap if nescessary.
Your problem really sounds like a power-rise-time issue. Compare the behavioure of the circuit when shutting off DC vs AC. Does it work properly when shutting off/reconnecting DC?
BTW I hope you're using the RESET out and not the /RESET out of the 707.