I have a 2000 Grand Marquis. Yesterday I turned the key and the starter didn’t crank. Everything else came on like it is supposed to. Radio, dash, warning lights, etc. After trying a few times, it started but the starter wouldn’t disengage. I had to disconnect the battery for it to stop. I changed the ignition switch and cycled the key, but when I try to hook the battery back up, the starter immediately begins to crank. Even when the key is off and out of the ignition. Ugh!
I would suspect either a stuck starter solenoid (mounted integrally on the starter assembly) or a faulty starter relay, located in the underhood fuse box, I think. See if your owner’s manual specifies which is the starter relay and unplug it to see if that helps.