That type of circuit should work fine if you add an inverter to the output (essentially giving you Q and -Q), but you are limited only to 2 circuits. 555 is nice and cheap and simple, and a 7406 is also cheap and simple.
If you need more than 2 seperate circuits, then I would suggest a binary counter, and multiplexor.
If you could use DC to drive the coils then you could use simple and cheap NPN power transistors or N-channel MOSFETs - though both need adequate clamping diodes to protect them from the back emf.
If you are using AC to drive the coils, then for the switches I would use solid state relays, easily built from a triac triggered by an opto-coupled triac (this will easily handle 240 V mains, depending on the quality of your construction) and an appropriate RC snubber. However, triacs have the complication that they don't automatically stop conducting - the current has to stop for another reason (with AC this happens at the end of a half-cycle, 120 times every second), but this would mean that for short periods both circuits could be energised.
Conventional relays could be used in either case as long as they are fast enough - in either case, you will probably need to use a transistor to drive them as the 7406 is unlikely to manage.
