What you want to do is find a lock cylinder that DOES turn with the key- it's possible, especially if you live in a state that uses salt in the winter for the roads, that the cylinder is stuck or seized, or it could be freed by spraying some WD-40 in there with the thin straw that the can comes with.
If any of the lock cylinders- driver, passenger (if yours has one here) or the trunk work, then disconnect the battery, put the key in that lock, reconnect the battery. and then turn the key in that lock to deactivate the alarm.
You could also try putting a new battery in your remote- if you flip the key out, you will see that there is a small groove where the two ends of the remote come together. Put a screwdriver in there and twist to pop the two halves apart. Then, the section with the buttons on it can be pried open along the edge, and the battery replaced.
Note that there is a small groove on the back of the remote, where a tab from the upper "key" portion of the remote snaps onto the bottom part. DO NOT pry here thinking that it is easier to release this tab- it will almost certainly break, and the only way to hold the key portion to the button will be to tape them together (as gluing it would keep you from changing the battery)