Psychopathy is not just about a lack of empathy, there is a lot more to it, such as a need to and get pleasure from looking at or causing suffering in others, often with wanting complete dominance, power and control over the other.

What do you mean by having false emotions, do you fake them with words and body language? If you do, why do you fake them? Is it because you don't feel them but still want to appear as a normal functioning person? It can also be that you don't feel much emotion yourself, but that your body still reacts on things, animals or people around you with emotional reactions, even if it's just minor like a smile when a person greets you in a friendly way.

Acting different in every single conversation can be a sign of adapting to much to another person. By looking at the reason why you're doing this, it could make it more easy to identify the problem. Some people may want to please every person, some want others to think of them in a positive way, some want to act in a way how they think others want them to act as a way of having "good" contact with the person, some act so much in "favor" of the other person that they forget they are a person as well with own wants, needs, interests, knowlegde, thoughts, feelings and opinings and some may act overly empathic, this could also explain why a person would act different in every single conversation.

Not having much emotions can be a result of abuse, such as narcissistic abuse, where a person is required to give up their own wants and feelings for the need for esteem of another person, but regardless of what the cause it, it doesn't have to mean a person also lacks empathy. A person can have barely any emotion, but still have empathy, this is very typical for people who dealt with narcissistic abuse.