I don't speak for everyone here but calling yourself an empath is simply a label. Some want to use it and some don't. Those who have known about their gifts longer tend to choose not to use it. It is a matter of choice. To me, from what you said, it sounds like you are at the very least a sensitive. That can mean all sorts of things, up to and including an empath. You have to look inside yourself and decide what you believe. No one can tell you what you are and what you aren't. You have to be the one who decides that in the end. If you are an empath your heart will show you the way. In the mean time, here are a link that may help you.
And just for reference, in case it is helpful...
These are the10 levels of the empath:
Psychometry- the empathic ability to receive energy, information and impressions from objects, photographs or places
Telepathy- the empathic ability to read people's thoughts
Mediumship- the empathic ability to feel the presence and energies of spirits
Physical Healing- the empathic ability to feel other people's physical symptoms in your own body (and often the ability to heal, transform or transmute them)
Emotional Healing- the empathic ability to feel another person's emotions
Animal Communication- the empathic ability to hear, feel and communicate with animals
Nature- the empathic ability to read, feel and communicate with nature and with plants
Geomancy- the empathic ability to read the energy of places and of the land - geomancers can feel the energies of the Earth, such as Ley lines. They can also get headaches, pain or anxiety before earthquakes or other disasters occur anywhere on the planet.
Precognition- the empathic ability to feel when something important is about to happen (often this can be a feeling of inexplicable dread or doom)
Claircognizance or Knowing- the empathic ability to feel what needs to be done in any given circumstance, often accompanied by a feeling of peace and calm, even in the midst of a crisis