Is your fireplace wood burning or gas? If it is gas, was it a wood burning masonry fireplace that was converted to gas or was it created as a gas unit from the start?

woodburning. House built in 1985

If it is woodburning then the flue should be able to be fully closed. You can do as you described. Of course if you burn a fire, then the flue must remain open for a long period after the fire goes out (like 8-12 hours) and you will then just be funneling the heat in your home out the chimney with no way to stop it.

Thanks. I never knew about the 12 hours AFTER!! That makes sense....

So, if the flue seems to be completely closed, but there is still a draft, should I have a chimney expert look at it for proper fit?

That would be a great idea. Wood burning fireplaces should be examined every few years anyway. I didn't intend to say the 12 hrs is some type of rule -but to indicate that the fire must be fully out and the area cool before closing the flue. My parents had one flame back up 24 hrs later. Fortunately someone was home and the damage was contained.