If you want to avoid 'write', the most natural thing to say in this context is this: 'Could you do the cards while I'm cooking dinner?' If the stamps, pen and address book are all there, it is absolutely obvious that someone needs to write the cards. So, in fact, there is no need to say 'write' at all - the all-purpose word 'do' ( indicating a job that needs doing) is enough. In a similar domestic situation, you might say 'Could you do the vegetables while I do the dishes?', as it's obvious from the context that the vegetables need cutting and the dishes need washing - there's no need to be any more precise. By the way, 'Will you please write' sounds awkward and over-formal between a husband and wife. You'd only use that tone with a partner if you are deliberately being cold and/or trying to stifle your annoyance. 'Can?' 'Could?' or just 'Will?' is OK. The 'please' isn't necessary, but if you do want to use 'please', you should put it at the end. It sounds less formal/angry than putting it in the middle. One last thing - we never use 'Mrs' or other titles on their own. You could say Mrs Brown, for example, or else just 'a woman' or 'a married woman'. I hope that's helpful.

Assuming the cards are already filled out (meaning they already contain content of some kind), then one could say: "Will you please sign the cards while I am cooking the dinner". Sign, in this instance, means to write one's name (specifically put a signature on paper), which is why I have said this would make sense if the card already has content and just needs an ending and a name. If the card is blank and requires content then it would make much more sense to say write as the card most likely needs content to serve its purpose. Let me know if this helps. I will add that Su.Ki. has a very well written response. In regards to the title 'Mrs' being used on its own Su.Ki. is very correct. Had you said "The missus", meaning the man's wife, then that would be acceptable although fairly informal.