There was a time when each of these animals was perfectly adapted to their environment. Whether it was due to climate change, a volcanic eruption, a meteor strike or the effects of human influences on their surroundings, some form of change proved to be too much for all of them.
Perhaps you're the apex predator of your surroundings. If you're highly aggressive and people tend to take a step back when you walk into the room, T-Rex might just be your perfect match. Should you mention that your reach is rather limited and you tend to scavenge off those who are weak around you, there will be little room left for doubt.
Maybe you're someone who just wants to be left alone and to live out your days in peace on a tropical island. If you'd fit right in with the characters in "Alice in Wonderland" and you have some highly unusual characteristics, the dodo is looking like the front-runner to be your extinct avatar.
Alternatively, it's possible that you're most known for your slow and steady pace and the way you tend to make people laugh, or cry, from pure joy at your presence. If this is starting to remind you of the way you move through the world, the giant sloth could be just the one we're looking for.
Let's get a move on before these options disappear from the planet forever.