Congrats, that looks like a rather recent model. Might even be an early laser type?
So, that's the usual way to determine it does PS2: The adapter is included. No point hoping for correct documentation from either the manufacturer or the retailer, just not gonna happen.
But, I still do respect Logitech. They tended to be rather conservative with their support for legacy standards. Just their device-base pairing is a bit of a mess and of course, build quality deteriorated like with anything else.
Quite a testament to Duracell that the AA did not leak. Cheap alkaline cells are perfectly capable of doing that on their very own after a few years. Some D-cells on a shelf did that here, actually making a noise by venting gases.
Is it one or two cells? If two, consider getting AA-AAA converters to reduce the weight. Or some AA Lithium cells. Expensive AF, but much lighter and last forever.
What model is that mouse? You should always include that around here so others may find it, maybe looking for the very same info.