I am not sure anxiety is inherited, I have a feeling it may be passed down from a parent's own history of dealing with their dysfunctional family environment. In other words, your mother most likely grew up in a dysfunctional home where she developed stress and anxiety and that got to a point where along with her also dealing with an alcoholic husband, she began to self medicate with alcohol.

Back in the 50's and 60's, while alcoholism was known about, there were a lot of psychological challenges that were not very well understood like they are now.

The important thing is you are learning now and looking to learn more and get support for yourself.