1989 Cadillac El Dorado starting problems

One common suspect is a warped starter solenoid housing. To verify that, check for 12 volts on the smaller solenoid terminal on the solenoid when a helper turns the ignition switch to "crank". If you have that voltage, the solenoid should be engaging. A jump-start can overcome that heat-related warpage due to the higher system voltage.

You can also remove the starter relay and jump the two terminals in the socket, or you can remove the cover, then stick the relay back in that way and squeeze the contact. If that makes the starter crank, we'll have to work on the ignition switch / neutral safety switch circuit.

Should I buy a new starter? Or what can you suggest

I listed two tests. One will verify if the starter solenoid is defective and one will show if the starter is okay and we have to look somewhere else. No professional would replace the solenoid or the entire starter without doing these tests first to rule out other causes. Too many people waste money on unneeded parts when they just replace random parts.