When being introduced to others you should begin looking them in the eye, while starting to smile, begin the dialogue of introducing yourself, while reaching to shake their hands and doing this simultaneously.
You are correct. There is one element that has more of an impact than great eye contact as well as the others aspects of the introduction, the handshake. Why; because it can affect how you perceive others (and how they perceive you) and how and if you wish to proceed with building a rapport with the other person.
When people shake hands, energy is exchanged. And this form of touching can promote trust and competence or it can create barriers of mistrust and incompetence. Think back to that moment. What impact of her handshake affected your perception of her? That is why you felt uneasy. Your perception might not have backed up your expectations and/or their qualifications of the team member. Or if you did not have any expectations it certainly put doubt in your mind about something.
Eye contact is important as well. But when being introduced to others, have a great smile, great eye contact, say the person’s name but be sure that you back it up with a good firm handshake!