From what you have told me I am more enclined to believe the problem is in your pipes and not the tank. If you can try a few tests, they may help to solve this mystery. First, shut the cold water supply to the tank off. Be careful not to scald yourself and open the T&P valve (temperature and pressure) on the upper right part of your tank with a clean bucket at the end to retrieve some water. Only let it out for a few seconds and click the valve closed. See if the water is rusty. With the cold water still off, open the boiler drain at the bottom of the tank, with a hose attached to it and do the same to see if the water is rusty.
I'm thinking that there is a possibility that there are either old steel pipes or old valves, maybe both, on the hot water system and that is causing the rusty water to develop. If there is a dead ended pipe or an unused branch of pipings, the rust will collect and when you turn the water on it gets flushed out.
Try what I've recommended and look around at the piping in the house and see if you can find anything like the circumstances that I've described.