How would I check the switches to confirm they're good? sorry, I'm a mechanical guy, not electrical.
Well, Mr. Cyborg, both the units complete an electrical path to the engine case as part of their normal function.
You haven't told me what test equipment you have available, so I'll assume you have a meter, that anyone can have for $10-20.
Disconnect the switch unit being tested. Set your meter to read Ohms, low end scale. Place one lead on an engine bolt and the other lead on the post you previously disconnected.
Your neutral switch should read zero or nearly so when the trans is in neutral. At all other gear selections it should read very high, or infinity. (Which is a pretty high number)
Your oil switch is just the opposite. When the engine is stopped and has no oil pressure, the meter should read Zero or a very low resistance. When the engine is running, the meter should show very high numbers on the scale.
Alternately, with your ignition switch on, connecting each of the wires that were previously connected to the switch, with a jumper wire, to an engine bolt, should make the indicator lamp go on. Technically, you can do this last test with the wires still connected to the switch. If you can get the lamps to light in this fashion, the sender switches are bad by inference.
Don't forget to lube those joints Mr. Mechanical guy!
Cheers,