When using frozen vegetables, even if I let them unfreeze overnight, I am finding that they then render quite a bit of water when cooking. How can I drain vegetables more effectively, especially when I need to use them right out of the freezer?
You can't drain them while they're frozen. If you need to use them quickly, consider thawing them in the microwave, then draining them.
As for how to drain them, it depends on the type of vegetable. Greens like spinach should be squeezed out. Chunky vegetables should be drained in a colander and then dried with a towel.
Some of the water you get may be coming from inside the vegetables, though, in which case it's still going to come out only when cooking. In that case, the only solution is not to overcrowd the pan.
Both fresh and frozen vegetables contain water in them that expels during the cooking proces.
For some types of vegetables, soaking in salt or salt water helps remove excess water (and pesticides as well) in them. An example would be cucumber salad. People usually soaked the whole cucumber in salt water or slice them up and sprinkle salt on them, before keeping them in the fridge overnight. Or else, you may end up with a watery cucumber salad.
Depending on what kind of vegetables you are cooking and how you cook it, this technique can also apply. You can sprinkle salt on the vegetables or soak them in salt water overnight (or at least 6-8 hours). Remove the salt and the excess water by flushing in running water before cooking them.