1999 Cadillac EL Dorado security theft problem

The theft system problem is likely relevant to the ignition cylinder. There are two small wires that carry the key resistance down through the column to the module. These wires are prone to breakage over time. The stability light is most likely unrelated.

Ideally, replacement of the ignition lock cylinder with some new keys, is the correct repair. There are some options as far as bypassing the theft system. If that is in your interest, you would have to research it. Just be aware that there are about 15 different key resistance values. You need to make sure you match the new key to the existing value, or the vehicle will still not start. For example, if you got a used column from a salvage yard, chances are you would not have the same value key. You would have to have a new key cut, with the pre-existing pellet value for it to work.