When the security system begins alarming soon after hooking up that new battery (or for any other alarm "trigger" condition), it will send power in simultaneous bursts to the headlight and horn relays. You should hear "honk.....honk.....honk...." as egads noted, but not a constant & uninterrupted blare.
First, I'm assuming that your battery is still disconnected due to the blaring horn. Keep it disconnected for now. It's no fun working under the hood with that horn operating. The following is based on egads' assumption that the horn relay is stuck (I'm in full agreement with his diagnosis).
Make sure the steering column stalk mounted headlight switch is in the "off" position.
That horn relay is in the underhood fuse box. After you open and secure your raised hood, look toward the passenger side of the engine bay, in the far corner. Unlatch and lift up the cover on that black plastic box.
There will be the usual fuses and some little plug-in modules, black and rectangularly shaped (those are relays). Starting at the fender side, and counting inward, the third one away from the fender is the horn relay (the first two, located closer to the fender, are headlight relays, one each for each headlight). If in doubt, look at the top of the fuse box cover, as it has a diagram.
Remove the horn relay, put it in your right pocket, and reconnect your battery. If the security system begins alarming, you should hear clicking of the headlight relays, and see the headlights flashing on, and off, and on, etc., until you reset the alarm by unlocking the doors with the key fob remote transmitter or by unlocking either front door with the key (or just wait 2 minutes, and the system auto resets). This is per the owner's manual (wish I'd known this for the first 3 times my kids killed a battery). Go ahead and reset the alarm system, since you don't need it in "alarming" mode for this check.
Remove the headlight relay next to the now-empty horn relay socket (that's the left-hand headlight), and plug it into the horn relay socket. Nothing should happen if it's only a stuck horn relay. Tap the steering wheel to beep the horn (it should work...this verifies the circuitry from the steering wheel to the horn relay is a-ok). If the horn blared upon inserting this headlight relay into the horn relay socket, you have a problem with wiring between the relay and steering wheel, or a steering wheel horn button is stuck.
Either way, remove the headlight relay you plugged into the horn relay socket, and drop it into your left pocket.
Now, remove the horn relay from your right pocket and insert it into the now-empty socket for the left-hand headlight.
If the horn relay you now plugged into the left-hand headlight socket is stuck (which I assume it is), the left-hand headlight (driver's side) will turn on and stay on. Assuming you have already reset the security system, the right-hand headlight (passenger's side) will remain off.
If the left-hand headlight came on, remove that stuck horn relay again, and drop it back into your empty right pocket (the left-hand headlight will stop illuminating). Remove the left-hand headlight relay from your left pocket and plug it back into its respective socket. If in doubt, the empty socket in question is the middle one of the three relay sockets. If in further doubt, use the diagram on the fusebox cover.
Now, do a quick function check of your headlights by turning them on then off using the stalk-mounted switch on the steering column.
If all went as written (and it should), go buy a replacement horn relay.
My apologies if I make it sound overly simple, but I've worked on cars a while, and I've misplaced similar-looking electrical components. Luckily, I've caught myself. Now I label things with masking tape and Sharpie markers. .
This verifies whether or not the relay is at fault. If this does not work, PM me
v/r, OF