Usually, you should use an MVC framework in favor of Servlets directly, but that's not your question, and I'm going to assume you have a good reason to be implementing your own Servlets. On to the answer...
Per this - https://www.hibernate.org/hib_docs/v3/api/org/hibernate/SessionFactory.html:
Implementors must be threadsafe.
and
SessionFactorys are immutable. The behaviour of a SessionFactory is controlled by properties supplied at configuration time. These properties are defined on Environment.
So it's OK to share an instance of SessionFactorys.
In fact, from my experience, your HibernateUtil approach is the better approach, as SessionFactory creation can be very expensive.