When you are getting low on hot water, is the hot water pipe at the water heater to hot to touch? It should be about 120º. Does the hot water seem fine for a few for awhile and then it cools down? Normally, when there seems not to be enough hot water, the lower heating element is not working with only the top element is. Only one element is on at a time. As the top of the water heater hits 120º, the upper thermostat switches power to the lower element. If the lower element is not working then you have lees than a 1/2 working tank. However having a new tank the problem is something else.
I suspect that the hot water is backing up through the cold water inlet and mixing on its way up..
Sorry, but this statement is not possible. New water heaters have heat traps that act like a check valve. Water is greatly restricted or not able to flow at all up the cold water pipe. What can happen with old water heaters the fill tube that brings cold water to the bottom of the tank breaks off. When hot water is drawn, cold water enters the top of the tank and is drawn up the hot water pipe cooling the water. This is not happening here.
Similar to what Breplum suggest. Turn off the cold water at all faucets including the washing machine. At the furthest sink, run the hot water until it is hot as expected from the WH. While the hot wate ris flowing open the cold water shut off and check if the hot water temperature changed. Do not open the cold water faucet. Do this at each fixture. Don't forget to do the same at the washing machine but all you need to do is just open the cold water shut off. Check the temperature at a near buy sink.