I'm not sure there is such a thing as a "synchronous DC motor". A synchronous motor (I understand) is one which runs from AC, and rotates IN STEP with the AC. For example, a synchronous motor running from your 60 Hz mains will rotate at some sub-multiple of 60 Hz (like 1800 or 3600 rpm). The speed is then fixed, and is pretty stable.
The speed of a DC motor depends on the voltage applied, and on the load. This makes it a bit tricky to control the speed accurately, as you've found (more load > motor gives up). Also, the torque produced varies with the current through the motor's windings. As you wind down the voltage to reduce the speed, it becomes weaker, and more susceptible to load variations.
It is possible to regulate a DC motor, but that would require a speed sensor and feedback circuit. I expect there are low-voltage synchronous motors out there, that can be driven by battery-powered circuitry, with its own accurate frequency reference.
Regarding battery life, try measuring the current to your motor. A small, alkaline 9V battery has a typical capcity of 500mA-hr. This means that (for low currents) it can supply 500 mA for about an hour. This is only a rough estimate. Also, if you stall the motor, the current will shoot up until you let it go.
Hope this helps
Regards
Edwin.