I had an amnio exactly at 37 weeks for exactly the same reason.
I did not find it frightening or painful. It was a little uncomfortable but overall an almost indifferent experience. I did find out that her lungs were inconclusive which meant that she was 'cooked' additional week. Her lungs were more than ready when she was born. She screamed like there was no tomorrow!
If you are a person who likes her husband around for every appointment, go ahead and get child care. But if not, have him drop you off and pick you up. I believe I had a friend with me.
Congratulations on the upcoming arrival of your little one.