When driving inductive loads with a DC resistance or DCR, there is a motor surge current on starting equal to the V/DCR. In efficient motors this can be as much as 8x the rated load current running at full speed.
It is wise to use separate supplies that give much lower ESR impedance along with the Bridge driver RdsOn than the DCR of the motor such that voltage sag is limited to some practical value such as 10% max.
THe reason we do this is to minimize the I^2R losses in the current loop and then considering total load power choose a % loss from the battery and switches to reduce self-heating.
Thus you need to determine the DCR of the motor and then choose a bridge driver RdsOn and choose a power supply that can deliver the peak current with x% voltage drop.
If you do a load test on your AA batteries you can expect a fairly high ESR in the 1 Ohm Range, while LiPo batteries are much lower and may be better suited to this.
You also need to consider the Amp-hour and C discharge rating of the battery under load to retain achieve the potential life charge cycle of the battery. These may be 1C, or C/10 for example.