I'm fairly new to electronics. I wasn't able to find anything addressing this on the Web but that may be because I don't know the search terms to use.

I'm trying to design a device that contains two 555 timers. When powered on, one timer will be used to generate a repeating beep every 5 seconds and the other will be used to turn the device off after 1 minute. It will be powered by a couple button cells which preferably won't be replaced for the lifetime of the device.

Should the 555s remain powered on (ie voltage applied to Vcc pin) when the device is off? This seems like the obvious choice to save battery, but I'm worried that there will be timing issues if Vcc and the input pulse (to initiate counting) arrive at roughly the same time to the 555. Should I add a delay IC to the input pulse? Or should I use 7555s, which draw less current, and leave them powered on? Are they significantly more expensive? How long could I expect the button cells to last with a 555 vs 7555?
[![schematic (shown with 555s powered at all times)][1]][1]


  [1]: https://i.sstatic.net/zrtjm.jpg