First of all, cranking up any electronic device to it's max volume is going to create distortion. If you have to crank up the volume to full, assuming you can understand it at full, you need a new aid. Second, to get rid of background noise you should use either a t-coil or an FM/Bluetooth system. Use headphones with higher t-coil setting, just like a phone. Use the FM system with mike off so you hear only the music. Finally, depending on your hearing loss, you need to understand that your brain is filling in the music(if you already know the songs). You are not hearing everything just like when you had normal hearing.
For years, I did the same thing. I cranked the volume on full. Once I figured out the distortion issue, music clarity increased. Depending on the hearing loss, you might get different results. But, one thing is certain, you can't get clear sound at max volume. Period.