Answers

What make & model is your new water heater?

I would recommend contacting the company that installed the water heater

I would start by performing a flow test at your shower head. Get a 5 gallon bucket and open your shower head full open and time how long it takes to fill bucket. If you fill the bucket say in 1 minute, this means you are emptying 50 gallon tank in less than 10 minutes. (Divide 50 by number of minutes) Keep in mind as you use hot water, it is replaced with cold water, so hot water temp is dropping from the moment you start. Secondly, you most likely temper your hot water with cold which then extends this time. In short this is just a guide but tells you how long it takes to empty tank. If this works out that your emptying bucket in shorter time than your shower, concider checking flow restrictor on your shower head. If this is not the case then call the installer of hot water heater.  

Yes, contact the installer. Hae them check it. Just from the sound, it could be anything from a bad heating element, malfunctioning thermostat, to being plumbed backward, to name a few. Either way somone qualified to diagnose and repaitr the problem needs to look at it.