Thing is a left hand when he answers Gomez' fake phone call instead of his usual right. When he delivers the mail near the end, he is right handed.
Around 23 minutes, when Morticia announces that the letter is from her mother, Gomez does a few stunts on the bars. But the stunt at a slower rate will show that it is a stuntman, and not John Astin.
After Gomez answers the door to let in the doctor, he goes to close the door, but his hands misses it completely. He right hand fumbles with trying to find it, and eventually grabs hold of the door.