Is it necessary for a legally married woman to change her signature after marriage?

Nope.

Have you seen how doctors sign? It's like a scribble, you can't tell what their name is by looking at it.

Most married women change their signature to avoid hassles. If their maiden name was Jane Smith and their married name becomes Jane Cooper, signing as J.Smith when your name is J.Cooper may look suspicious but as long as there's this same exact signature on your photographic document (passport, identity card etc), then they will have to accept it.

So you can keep your signature for sure but it has to be consistent on all documents.