There are many different types of people. People like you are the carers, the supporters. It took me a long time to learn that in order to care for others you have to take care of yourself first (easier said than done and I still struggle). There is a reason we desire to be selfish. Some give into that desire more easily than others, but you shouldn't completely deny it yourself.

Make sure you look after yourself first. Also, there is not necessarily as much shame as you might think in asking those close to you for help. Sometimes our pride can bring us to harm.