Did you notice the two spring-loaded horn contacts in the turn signal switch?  They may be compressed so far down they are contacting the housing causing a dead short.  Try this first, before you take it apart again...  Loosen the two 1/2" hex nuts that hold the steering column to the bottom of the dashboard.  Don't remove them, just loosen.  Then slide the column tube down towards the steering box a little bit.  You want to open up that gap between the steering column housing and the new steering wheel adapter, to give those contacts room to move.  If that works, don't forget to rtighten the nuts.

If that doesn't give some relief, then you have to pull the wheel and inspect the switch and figure out what all got hosed up during the installation.  Once you pull the wheel, take some good digital pictures and post them here so we can see what you have done.  NOTE:  You have to make a certain number of posts before you can post a picture, so if you have any other stuff you might have questions about, now is your chance!

Also... Welcome to FYIFORD.  You are going to like it here.

I'm ready to scream!!! Counting this time, I've had the steering wheel off 5 times now. I slid the steering column down but it didn't look like those two contacts were compressed. I now have the wheel off again and when I hook the battery up, the flasher under the middle of the dash still clicks with the key off. Still no indicator lights or turn signals. It would make sense that the new steering wheel installation caused this but I have no idea where to go now.

Thank you. Just found a new thing to add to the plot. The clicking was actually activating the right rear back up light. I replaced the wiring harness to the rear of the car. I did a function check and everything worked. The last electrical I did was to hookup the new backup lights. So I guess somehow I got a short somewhere? I have unhooked the back up lights and that stopped the clicking. However, still no turn signals...

Okay. I have enough patience to try this again. Rebuilt my 66 coupe. All new wiring harness from firewall forward. Factory coded so no big deal. Worked fine. All electrical was working fine. Disassembled for paint. Completed paint. Added new front parking/turn signals. Still had bulbs in back. Checked electrical; it all worked. Put in new wiring harness to rear of car. Functioned perfectly. Have not hooked up new back up lights. Hooked up new back up lights. Still seems pretty simple. Checked them, didn't work. I didn't think too much about it then. Put aftermarket steering wheel on. Instructions said take both springs off steering shaft. That was a mistake. The inside one held the sleeve down keep steering shaft from wobbling. decided to try turn signals. Nothing. No indicator light nothing. Moved mustang and noticed horn would come on when steering wheel is moved just so. Put inside spring back in and no horn and no turn signals. Took a look at fuses. Noticed bottom horizontal fuse blown. Replaced and suddenly hearing flasher clicking about mid-dash. Pulled out a rusty cracked one. Replaced that one and one by steering column with a 536. still clicking. friend noticed the right rear back up light flashing in time with flasher. Now the fuse blows in less than 2 seconds when putting a new one in. Unhooked the 6 pin connector and put in new fuse. A few clicks and it blew. Looked at three pin connector from rear of car. Noticed the single pin on it was black w/red stripe. Same as back up light. had a blue hot wire connected to it from under dash. Disconnected the blue wire and put in new fuse. It didn't blow!! Now I have the back up lights disconnected and still can't figure out why I have no turn signals.

From one 66 coupe owner to another, you're preaching to the choir on steering wheel issues.  AFter replacing the entire signal and horn unit one of the contact got stuck while driving.  Essentially anytime I moved the steering wheel more than a few degrees either way the horn would go off.  Sounds funny now, not so funny in rush hour.  I like the look of the original wheel but it's just so big.  W/o out a tilt column and limited seat adjustment it kind of makes it a pain.

I enjoyed a similar problem and had many helpers from this forum.  See my thread listed as 65 FB Lost my headlights.

My problem was different than yours but there was a lot of info given to me to help resolve and I think you may be able to use this info in ferreting out your issue. Good luck.