The neutral safety switch is located at the bottom of the column. I would suggest that you hold off on the starter, because I don't believe that is the problem, until you check all the rest of the items. First start by turning on the headlamps, and seeing if they are bright. If not bright, then look for a poor negative or positive battery connection. Turn off headlamps, and then with the key turned to the start position, pull the shift lever from Park to the extreme right of the gear selector. If you hear the starter start and then stop, do it more slowly, until you find the "sweet spot" where the car will start. This would indicate a bad neutral safety switch. If this test doesn't give you positive results, then the next thing to do is to ferret out the neutral safety switch, and see if it has come apart or come loose. The Cadillac neutral safety switch has a multiple myriad of functions, such as providing vacuum to the parking brake, turning on and off the backup lamps, etc. Finally, I would give a tug on the positive battery cable where it goes down toward the starter. Sometimes this will give you a temporary fix to a poor connection. Below, thanks to eBay, are some pictures of the neutral safety switch. Be careful not to break the vacuum line connections, because those small nipples are very brittle from age.
If none of the above produce any positive results, then it might be the starter, however, rarely do starters catastrophically fail. Usually they will give you some warning. If you have another car available, and jumper cables, give that a try. I have seen where a battery will have one cell fail, and there will not be enough reserve voltage to even get the starter solenoid to click. This will show if you find the headlamps dim when you attempt to start the car, as directed above.