Skewer removal without quick release
You have the wrong tool. Your cassette lockring removal tool has a long pin emerging from its centre, which would fit inside the axle of a QR compatible wheel, to keep the tool centred.

What you have named a skewer is actually the axle and yours is solid, so the axle blocks the tool. The correct and simple way to remove your cassette is to get an inexpensive cassette lockring removal tool which is compatible with your wheel, something like this:



Or possibly one with a handle already attached if you don't have adjustable spanner or sockets to suit the size of the tool.

If you start disassembling the wheel axle by removing locknuts, you will have a much larger task which typically requires further tools (cone spanners) and extra skills to accomplish and your tool will still not fit without removing the axle completely. Those skills are worthy to learn, but unnecessary for simply replacing a cassette.