How do I keep my zoodles from being too watery?

Here are a couple of options I've found online:

Dry them in the oven

Dry the noodles out in your oven at 200 degrees, for about 10 minutes or until they are finished “sweating.” You’ll want them dry to the touch, but still very flexible and moist on the inside. Stir them after about 5 minutes and see how they are doing…the time they need to dry will vary depending on how moist your squash is. Once they’re fairly dry, transfer them to boiling water for just a few minutes. I stood over the pot and watched as they turned darker green, it took about 4 minutes for my noodles to reach a texture that was similar to al-dente.

Encourage them to "sweat" out the water by using salt:

Sweating the zucchini noodles with salt is essential in this Paleo recipe so they don’t become watery and limp during cooking.
Place the julienned zucchini in a colander or wire strainer and toss generously with salt until the strands are lightly coated. Allow the zucchini to sit for 20 to 30 minutes. Rinse with running water, drain well, and squeeze dry in a clean dish towel.

Optimal texture is about optimal heat transfer. When a vegetable releases so much water that it simmers in its own juices instead of pan-roasting, you should increase the heat transfer.

To achieve that, you need to 1) use high heat, 2) not overcrowd the pan, and 3) use a pan which is capable of better heating. Cast iron is perfect for these situations, heavy steel pans too. Aluminium is much harder to work with, especially if it's thin. Also don't use a nonstick pan, you'll destroy the coating at the temperatures needed.

This technique can be used on its own, or combined with something else, for example oven drying. But note that a zucchini is over 90% water, so you won't be able to simply get it dry enough to resemble a dough noodle.