Chiming sound when I take the key out of the ignition

Your problem sounds like the key buzzer is sticking in the ignition cylinder housing. You could try lubricating the cylinder, spray some wd-40 or silcone and hope it makes it to the buzzer. Otherwise, the cylinder needs to be removed and inspected.

Just to add to this one is it the slow chime beeping or the fast one like you left your light's on?If it's the slow one then when it's beeping and you have the key out of the ignition see if the key cylinder is still stuck down like when you have the key turned on. If it's not then you have a stuck key minder button on the ignition switch. You would have to replace the ignition switch if that was the case. The most common thing to happen on those cars is the key minder button to stick on the ignition switch. Let me know what you find.

So is the ignition cylinder sticking in the down position when you remove the key?Let me know so I can get you the correct instruction's for replacing the cylinder or the switch. The cylinder is a lot easier to change then the switch.

If it's sticking down then most likely there is probably brass from the tumblers wearing making the cylinder stick. You can try lubing it or taking it apart to clean and lube it or just replace it. If you just replace it and go aftermarket then all the one's I have seen are already keyed so you would have two key's one for the ignition and one for the rest of the car. Unless you got a lock smith to code the cylinder to the correct key code.I would try cleaning and lubing it if it was my car before I replaced it. Here is how you take it out.
1. Disconnect the negative battery cable.
2. Remove the lock cylinder bezel from the steering column shroud by carefully prying with a thin bladed screwdriver.
3. Remove the shroud fasteners and remove the steering column upper and lower shroud panels.
4. Insert the key into the ignition lock cylinder and rotate to the RUN position. Then rotate the ignition cylinder back to the ACC position.

5. Depress the locking button on the side of ignition module assembly with a thin pick-type tool. Slide the lock cylinder assembly from the ignition module.

Installation Procedure

Important: The actuator blade within the ignition module housing must be in the ACC position to install the lock cylinder assembly. If rotated, replace it to the ACC position using needle nose pliers.

1. Insert the key into the ignition lock cylinder and install into the ignition module.

2. Install the steering column upper and lower shroud panels and the shroud fasteners.
3. Install the lock cylinder bezel.
4. Reconnect the negative battery cable.

I like to clean them with some brake cleaner but you will have to seperate the cylinder with the tumblers from the little housing that come's out when you remove it like I gave you instruction's for. On the side that goes back into the big ignition housing you removed the cylinder from you will see a spring loaded pin that keeps the core from coming out. Move the key till that pin come's close to the long channel then push the pin in with a pocket screwdriver and keep turning the key towards that channel until the pin come's out. Be careful not to loose the pin or spring it can shoot out. Then remove the core clean everything with brake cleaner and I like to lube up with some graphite for lock's. Let me know how it goes.