as far as i know, only distortion/fuzz pedals and maybe ringmodulators sound different with a new or used battery.

if you like the tone of lets say the BMwTW with a new battery, there will be imho no audible change in sound if you use an adapter.

Analogman though has to say something on the topic and recommends the great idea: the virtual battery from gigrig!

analogman writes on his site:
A power jack may add AC hum noise, and a power supply can affect the tone, as the best sound is obtained by using an old style non-alkaline battery. I get these two for a dollar at the DOLLAR STORES. I like the Maxell or Panasonics. An Alkaline battery will sound a bit dull in a germanium fuzzface, it can lose some character (you can hear the difference in a sound sample above). It’s not just the voltage that makes batteries and power supplies sound different. It’s the resistance, inductance, and capacitance in the battery, which is part of the circuit in a fuzzface.

A power jack can also cause oscillation (squealing) on some high gain sunfaces with some power supplies. TheGigRig virtual battery should solve oscillation problems, along with isolating the power supply so you can use any power supply with our positive ground pedals. The VBATT also solves ground loop problems which occur when you use a daisy chain or the same power with more than one pedal. We now offer the Virtual Battery as an accessory and sunface option on our shopping website. Here is more info on the GigRig Virtual Battery

br
diego

I recently got an AC adapter for my BMwTW. Previously I had been using batteries for it. The AC adapter is the way to go. It makes it sound like it has a brand new battery all the time. Very full sound.

The batteries with the cat on them fade too. Try them on a compressor pedal.