The first thing to do is look at Millers web site and download/get a copy of the manual for your machine, and print it out. Do a search for Wildcat 350D, but if it doesn't come up that way go to nearly the bottom of the list because that's where it's listed alphabetically (Wildcat)-vs- numerically (350D). I've got one of those machines with the Waukesha diesel so I know the setup your looking at. Too there are full wiring diagrams in the manual.
That said, the chances you damaged the engine with the ether are minimal give the situation you describe, so I really wouldn't worry about that. Not saying it can't happen but no likely unless you kept cranking on it and it got to knocking from being over ethered.
That said the sending unit, the one that turns on the idiot light, is a part of the safety shutdown system. As such it works in conjunction with the rest of the Murphy shutdown system to protect the engine should it lose oil pressure or overheat. That being the case the switch, according to the wiring diagram should be a normally closed switch that will open when oil pressure is supplied to it, and close to ground when oil pressure drops. This causes the shutdown system to drop power to the injection pump and shutdown the engine.
Basically what I'm telling you is that you don't want to replace a system that automatically protects the engine wether you are watching it or not, with a set of guages that will allow the engine to over heat, run with no oil pressure, etc while your at the other end of a 50 foot lead and non the wiser that your engine is on it's death bed.
Beyond that many 'senders' as your calling it (it's actually a switch) are designed to be NO vs NC. If the one you got was NC vs NO, it would shift to the wrong position when you started the engine and cause both the light to stay on and the shutdown system to not send power to the injection pump...so it will not run... Now,if the old switch is making a little, but not quite enough it could possibly cause the system to send enough voltage to the injection pump to turn it "ON" but not necessarily enough to turn it all the way "ON". Too there is a fuse in with the over ride button and it may be partially blown causing the same problem.
Ultimately if I was there it doesn't sound like the problem your having is too complicated, but trying to tell you evertything to look at, and how to do it online like this is near to impossible. Your best bet, again, will be to get to Millers site and get a copy of the manual for your machine, with a diagram that will tell you evetything you need to know about the electrical system/shutdown system that you need to know.
Below is a link to their site and the page to get the manual. Good luck and post again if you've got any more questions and/or info that would help troubleshoot remotely.