It's possible that the rice is prewashed. I mean, for one, they don't sell anything called "sushi rice" in Japan, and secondly, for example in Viet Nam "sticky rice" is especially sticky rice (mochi-gome) that you wouldn't usually use for sushi. So clearly, you have some kind of exotic rice that is unique to your country or market or what not.
If it's pre-washed, not only do you not need to wash it but washing it may ruin its taste (they sell "musenmai" pre-washed rice in Japan and it tastes bad if you wash it at home).
But the most important thing is to let it sit WITH THE LID STILL ON after you turn off the heat. This is what makes the rice moist and, in your sense, sticky. Any kind of Japanese rice will end up gritty if you don't let it sit at the end.
Also, typically, Japanese style rice is cooked with the lid strictly on, so if you haven't tried that, try keeping the lid on at all times while boiling.
If you want to make vinigrated sushi rice, season the rice after you let it sit but while it's still hot. Here is a movie showing an easy way to do it.http://www.youtube.com/watch?v=-QzNDwnUQ2Q
As long as you keep the lid on the cooked rice, the rice will stay moist and sticky enough while it's warm. But once the cooked rice gets cold, it will get gritty. That's how it is.
You can still utilize this cold rice by stir-frying it, by stewing it, or by sprinkling water on it and re-heat it in the micro-wave. Otherwise, make an unseasoned rice ball while the rice is still warm, wrap the ball with saran wrap and put it in the freezer. You can micro-wave the wrapped ball to enjoy its original moist feeling again.