Just relax about the car.. It's a looker and you yourself said very well cared for. VERY few problems have gone unresolved here or ended up having to go to the dealer for repairs. And FYI .. many memebers are willing to personally help out. Adding a location to your profile is really a good idea.. And as noted, if your really not sure what your dealing with, pictures will go a long way.
Oh, and welcome to the forum.. we really are a good group here, some just get a little testy with winter..

 

1. you said "under the hood"
2. can you SEE anything moving? ( like the radiator fan )
3. if you can't see anything, is the smoke coming from the front of the engine ?
4. remove the engine cover ( pull up at the rear to unsnap from the hold-downs, then carefully at the front on both sides so as not to break the plastic. The whole thing will come off along with the 2 plastic intake pipes )
5. is the smoke coming from the round looking thing mounted on the front of the engine ?
6. If so, then you most likely have a stuck relay for the air injection fan in the RCM. - search for "Mystery No Start" to see about the RCM.

 

You can click the "Video RCM" in my signature to get there..

 

Here is the picture of the problem area my hubby says. We would like this fixed so we can keep it, as we do like it. We are from Alabama. Someone asked where we are from so I wanted to answer that too!

 

I lightened it up a bit.

smogfan_zpsa6a4c287.jpg

 

Yep, that's the air injection pump.
It SHOULD run for about 30-45 seconds after first starting the engine when it's cold.
If it is running all the time, then most likely the relay contacts for the pump in the RCM are welded.

( this is fixable, usually )

Meanwhile, just pull the plug connector off the pump, you might get a CEL but nothing will be harmed.
( it's sole purpose in life is to pre-heat the O2 sensors on a cold start )

( I too am in Alabama, just heard that Gus is coming back to coach Auburn next year. WAR EAGLE ! )

 

After it stayed running, I unplugged it and left it off for a full day and it has not come back on yet.

We are Bama fans lol maybe I shouldn't say that if you are helping fix my car!

 

Why not ? I hear they have a right decent team this year :-)