"Just the Way You Look Tonight" is pretty common for Fox Trot, and is used in studios quite a bit when teaching that dance.
"Blue Moon" would be a little slower, but still a good Fox Trot. The same with "Summer Wind."
"Under My Skin" works too, I believe. I don't recall dancing to "Fly Me to the Moon" in some time, so I can't say for sure on that one.
Oddly enough, "I Won't Dance" which is really a cool song, is a little tougher because its tempo is not the best for Fox Trot.
If you want a similar crooner with great Fox Trot tunes, Bobby Darin's "I'm Beginning to See the Light" and "Beyond the Sea" are lots of fun for Fox Trots.
But you absolutely cannot go wrong with Sinatra. He is the greatest.