The main point of OAuth is to allow a third-party to use your API on behalf of a user. It basically ensures that the user doesn't need to share their username/password with the third-party.
If you are not going to allow third-party websites to connect to your website then there are simpler ways. It really depends on what your API is about, how it is accessed by the users (website, mobile application, etc.).
Sending the username and password with each request doesn't seem like a good idea, even over HTTPS, so maybe a better solution would be to have a "/login" function which returns a token on success. You can then use this token to access the API. The advantage is that you can expire the token in case of problem.