Introducing yourself first at a party is a small risk with surprisingly good odds.

Most people are relieved when someone else goes first, because they were standing there hoping somebody would. That is the part the dread leaves out. The actual rejection rate is small, and when it does happen the cost is a few awkward seconds, which you will have forgotten by the time you get home. The odds and the fear are badly matched here.

Do it twice this week rather than perfectly once. Your name plus one question is enough of a script. Ask something about them rather than offering more facts about yourself.