2000 Dodge Intrepid ES here.

We've been having issues starting the car for some time now. Sometimes it would crank a bit and then die, other times it would tick.
Last week the battery gave up, so we replaced it. The starting issues are still hanging around, though. This morning I got a crank and then nothing, some ticking, and finally, when I finessed the key a bit (pressed it lightly and avoided the end of its travel) she started up fine.

Today on my way home, she did something new.

I turned on the car - started right up.
Heard a motor under the hood that wasn't normal, so I turned the car back off.
The engine died, but I recognized the sound as the starter still spinning!
If I turned the key on, the engine would start and the starter would continue.
I pulled the key out completely (it didn't want to release, but did eventually) and the starter was still going!
I popped the hood looking for a way to disrupt the starter. Didn't even reach the relay box by the time it decided to stop on its own.

Needless to say, I was looking a bit like the RCA dog...

I'm thinking...
- Ignition Switch
or
- Relay

Any thoughts?
Many thanks!

 

Quick related question, too...

If this happens again, and I can't seem to get it to stop by conventional means... Would it be super-horrible to pull the starter relay in order to kill it, or would I be endangering myself, the car... the universe, etc???
It's not like dividing by zero, right?