A battery charger puts out rectified DC with no filtering, so it has a lot of ripple. It also has very poor voltage regulation; in fact it is more a current-source rather than a voltage source. No wonder your motors are not well behaved.
If you can lay your hands on a 5 to 10Ah 6V battery, try that as a temporary power source for the motors. The motors may run much better in both directions.
In any case, you should be looking for a 5 to 7Vdc, adjustable, electronically regulated, 10A power supply. Check the surplus stores. They frequently sell industrial power supplies like that. Most have a screwdriver adjustment over that voltage range.