As shown in the pic (attached), I had already tried these two methods.

Everything was okay in both setups, but it still functioned like the old switch. The top switch operated both at the same time, the bottom did nothing. Does that help any? Does the fact that the two terminals are bridged affect anything?

I've wired outlets where one of the two is controlled by a switch, and that bridge had to be broken. I think in the case of a switch, that tab is broken if there are two different power sources.