Powering a load with specified maximum current of 300mA. Is it safe to use a matching voltage power supply with a higher current rating?
The answer to your primary question is covered completely by Olin Lathrop's answer in Choosing power supply, how to get the voltage and current ratings?

My answer is going to address your second paragraph about the 9V battery. You are correct about surmising the circuit was trying to draw more current than the battery could supply. A typical store-bought 9V battery is actually pretty awful at supplying anything more than a few 10s of milliamps without its voltage sagging significantly below 9V. 

You are also correct that increasing the volume on your amp pulls more current from the supply. In its simplest form, a volume knob can be just a variable resistor. As you turn the knob to increase the volume, you're actually decreasing the resistance in series with the audio signal to the speakers (in modern electronics, its not that simple, but good enough for a description here). Less resistance means more current, which means louder sound.

But Ohm's Law is absolute. With the volume turned all the way up, the effective resistance of the load is a (nearly) constant value. And power supply voltage is (nearly) constant as well, so current must stay constant. The current is effectively capped on the upper side no matter how beefy the power supply is.