How do I handle an extra black wire when replacing dimmer switches?
The wire that connected to both switches is likely the line (power source). One circuit and cable comes into the electrical box with a hot and neutral. The hot is connected to both switches, and then each switch has its own cable to the circuits they control.  If you remove it, you likely won't have any power to your fixtures.  The configuration will be the same regardless of if its a switch or dimmer.