Need help, possible head gaskets

So, new here. Searched around and couldn't find exactly what I'm looking here.
Have an 05 gc with 3.7. The heater core has gone out, twice. Never hit the HOT side but it did get hot, both times.
I have oil loss, and coolant loss. Neither on the ground. Oil seems clear of coolant. No coffee coloring or white "bubbles ". Coolant doesn't show sings of oil.
I'm thinking head gaskets are shot. It dies smoke pretty good, with a weird smell, when starting. After say 10 minutes, smoking clears. Sitting at red lights a notice smoke/steam, just briefly.
Also have to replace the conductor plate( seems straight forward).
My sons turning 16 next year so figured we would fix this. Other than head gaskets what should I look for?
We are planning on getting the heads rebuilt . Heard the stock ones had issues.

 

We had the same issue with my sons LJ. Had been running good. Got stuck in Atlanta traffic one day on the way to Athens. It never overheated but it did get a little hot. After that incident coolant started disappearing. Lots of white smoke on startup that would go away after 10 minutes or so. No leaks in the engine bay or on the ground. I bought an upgraded new head from Clearwater Cylinder head in Florida. The head was really nice and fit perfectly. Got a little HP bump with it. No leaks and running good. Coolant is maintaining now. So I assume my head was warped. Had it rebuilt by a local machine shop and sold it. It Could have just been the gasket. But by the way the coolant appeared to be seeping into the engine it seemed like a warped head.
I hope yours is just a gasket and all is well after a gasket job.

 

Our oil did have coolant in it. And we had brown sludge in the radiator.

 

So if you don’t mind possibly having to remove the head twice I would start with replacing the gaskets first. If you are rebuilding the head at the same time I think there is a good chance that will resolve your problem.