Hello: To start, there should be no air in the top of the tank. That suggests the pressure is indeed falling below zero, but a gauge would be nice, to know for certain.
If you install check valves you really do need to install expansion tanks or the hot systems will be subjected to pressure from thermal expansion up to 150 psi, which is when the relief valves on the heaters will pop. Tha sort of operation will ruin the relief valve and stress piping and tank/s.
I know nothing about the vacuum breaker that comes with the tanks. Perhaps David can help there, or Rheem should be able to.
By the way, it’s a good wierd to be listening in on your heaters. Wish I’d thought of a stethoscope! 😎