Savanna is a very common biome in Minecraft; you really should have little difficulty finding one if you explore a bit. Not finding on 3 or 4 attempts is simply bad luck but well within probability. How much exploring did you do beyond your spawn point? Perhaps you were looking for something more specific, like a Savanna village or the rarer Shattered Savanna biome?
No matter. Here's a map I found playing around with Amidst the other day. You spawn in a small patch of Forest biome, surrounded by Plains and a small patch of Sunflower. But as the map clearly shows, there is copious Savanna nearby, including a large patch of Shattered terrain and two villages. What's more, there are Flower Forests to the East and North and large expanse of Mesa, including a sizable patch of rare Eroded Badlands biome.