I'm not familiar with your particular setup or having a machine with a pushbutton HST selector as mine are manual.
Can you find the wiring diagram online or in the Parts or Shop Manual? Good to have and this may be a reason to obtain same.
You indicate the switch has 5 pins. One would be power in, one probably is ground. You might be able to figure this out by back-tracing your wiring harness to a connection and / or by wire color. Then the three remaining would be the H, M, L settings you mention.
Sit down with the switch, check readings between pins and write down the ohm readings. I suspect you'll see a pattern; and if you don't you've confirmed a bad switch.
Depending on age of the unit, I'd think a bad switch would be a low probability. Possible, but borderline unlikely. Switches like this are designed with a life of 250,000+ cycles. Unless there is weather intrusion or physical damage or corrosion on a pin or the socket.
I'd also suggest you use an analog-type ohmmeter like Simpson (top line) or Harbor Freight / Radio Shack / Lowes instead of a digital. You're not looking for a highly specific number of ohms, but rather full scale, 3/4-scale, half scale, 1/4-scale, that kind of thing. A digital meter might drive you crazy flickering between digits and if an area of flourescent lighting might give you strange values.
Please post back your experiences and what you learn so we may all gain knowledge.