How can I cope with the overwhelming feeling of being a new parent?

Find support. Find support from friends and family. There are online sites for new parents and support groups in almost every town. Be honest about what you need, even if it is time alone or time to sleep. Keep in mind that everyone feels overwhelmed in the beginning. You are not along.
Talk about it. It is very normal to feel overwhelmed as a new parent. Try reading any number of new parent books, talk to other parents you know and get support. Just admitting to being overwhelmed can help. Get a babysitter and get out of the house - take time for self-care as that is critical. Exercise, get rest (which is very hard but helps soooo much). Ask others to help you out and talk to your doctor.