How to prevent pressure from dropping from my water heater?
It appears to me you are building pressure using the bladder tanks. Once the pressure reserved in those tanks is depleted your pressure drops because the flow is restricted (possibly ahead of those tanks). I would agree with Justin that you have a restriction that is reducing the flow and after the bladder tanks are depleted you are only getting the flow through the main line. 

The fact that it takes longer to fill the sink with pure cold water tells me your problem is a restriction in the line feeding the water heaters and the bladder tanks are compensating for this somewhat when you fill the sink with pure hot water.

Look for a restriction in your main cold water feed line, and then at other points in the system. Check the air pressure in your bladder tanks with the line drained. Lastly, the piping looks too small to me. I would upsize that piping for better flow.