If you are his main caregiver, are you able to take him to a see his doctor and discretely ask doc to perform a cognitive test? This will help everyone come to an understanding and it is a good baseline measure. Also, you can video the other behaviors that are upsetting and show your family. If your father is putting on an act, he definitely won't want to be videoed. If he has dementia, it may not matter to him that he is on record. At any rate, you and your family need to have a plan as he will only get worse and caregiving is strenuous in time commitment, physical exertion and financial output.
How old are you? Is your extended family expecting you to continue to care for your dad as he gets worse? If they don't help in significant ways, are they ready and willing to support you financially? Now is the time to decide boundaries for yourself or else their unrealistic expectations will lead you to burnout. You are a noble person for caring for your dad to this point. Make sure you go into the future with your eyes wide open and taking care of yourself most of all. Good luck!
Have you considered having your father visit the other family members, say, for Sunday dinner? If they see first hand how he acts, maybe they will believe you and pitch in.
It is so strange, and yet I've seen that also with my father. He can appear almost drugged and spaced out and slumped in a chair when I (or my brother) visit which makes me and my brother worry terribly that something isn't right - and sends me into a spin of worry and subsequent conversations with my brother about what we should do - and then if a visitor (friend) arrives to visit him, father comes alive , animated and able to have have coherent and complex conversations and the friends and extended family of cousins think he is "normal" and "absolutely fantastic for his age". This makes me think there is a psychological component to what is going on! Definitely talk to his or your doctor about his sexual talk.
How old are you? Is your extended family expecting you to continue to care for your dad as he gets worse? If they don't help in significant ways, are they ready and willing to support you financially? Now is the time to decide boundaries for yourself or else their unrealistic expectations will lead you to burnout. You are a noble person for caring for your dad to this point. Make sure you go into the future with your eyes wide open and taking care of yourself most of all. Good luck!