I know someone who is also an introvert. The sad thing is that people think she is a snob or unfriendly and doesn't want to talk to anyone. If you talk to her one-on-one, she is fine. But in a group setting, she will not approach a group, or join the conversation. People often ask me, "what's with her?" The more you avoid people, the more you are telling them you don't think they are worth your time.
To keep from being thought of in a negative way, be sure to ask others questions if you are in a social gathering. Ask about their job or family, where they live, how long they have lived in TX, where they grew up, etc. Have several questions ready. It just takes practice. The more you do it, the more you will realize that most people love talking about themselves and will enjoy your interest.
If it's a group of moms, ask if you can join them. It is the easiest way to let them know you'd like to join them. If you don't join in the conversation, they will wonder what they said to turn you off.
Stop assuming people don't want you to join them, whether there's a project or personal conversation, etc. (You can tell if it's really personal if they are talking very quietly with everyone leaning toward each other.) Stop thinking that you have nothing of value to add to a conversation. The few times someone is cold to you, assume the best. They might not be feeling well, they might have just heard some bad news, they might be tired, etc. Don't assume it's because of you. If you come across someone who really is cold, don't assume the majority of the rest of the world is like them.
Just keep the focus off yourself and you will be fine. The more you focus on yourself, the worse you will feel about yourself.