No, I use handkerchiefs all the time. Normally I use one a day (if I don't use it I keep it for the next day).
I usually keep 2 - 1 in each pocket, so that I normally have a clean one to dry my hands when using a public loo, or anything else where it might be handy.
If barely used, it goes in the ordinary wash.
When either of us has a cold, I put a bucket with a splash of disinfectant near the washing machine, and in go the hankies.
I wouldn't say that I never use tissues - if I absolutely have to go out when I've got a cold, I might take some, but then have to take a bag to put the used ones in!