Hi JesseJJ, welcome to the Forum.
I also had that idea a while back but decided to build something else so I dumped it. However, I think it should be possible to put the different secondaries on a switch as you did. It's normal that your DMM told you continuity between the different taps as they're basically just a wire wound over a magnet. So they'll just have a few Ohms of DC resistance. BUT for an AC signal there is an impedance which is way higher.
You want to make sure that the unused tap is really an "open" end in the respective switch position. And don't switch while running the amp!