Can you look at the blower motor?

Typically on a swamp cooler you first turn on the pump by itself for a few minutes to wet the pads before you turn on the blower. Sometimes when cooler, like in the morning, you may wish to just run the blower without wetting the pads and the pump running.

From your description of how it appears things may have been wired, it doesn't seem to provide for these rather normal set of options.

But since designs can vary, I think it would be best to look up the installation manual online for the recommended wiring on your particular cooler.

I just read your post again and in regards to the existing wiring, it looks like you may have the necessary wiring in place, although of course the grounds are still an issue.

If the black & white is your line side, the switched load side maybe something like red/black for low/high on the blower and orange for the pump, with the white neutrals tied together in the box. BUT, you need to verify this at the cooler first!

Don't know what the current switches look like but below is an example of a single gang swamp cooler replacement switch available at Home Depot.

Here is a link that might be useful: Replacement swamp cooler wall switch at HD