Easy way to test if the speedo drive is shifting, is simply shift the two speed under load up and down in any forward gear. The speedo drive will shift even though the axle can't. There will be noticeable change in the speed indicated if it is working.
It is possible for one to shift and not the other, as in the drive and the axle. One of my trucks has developed a quirk where in cold weather when ever I shift into one of the forward gears in the pattern, the drive will shift but the truck doesn't. When I shift again, the speedo returns to normal. I haven't tracked it down yet, but if I pull on the cable running down the shifter, I can make it do it intermittently. First place I will be checking on mine is the switch.
As said, 10.00 20s are the same as 11 22.5s. I run them mixed on various farm trailers.