Jeff, I think you should just go with your gut. Don't overthink things. If you walk into a room and feel like saying hi, do it. If you feel like touching her shoulder, do it.
Dont worry about what her reaction might be. Just go for it. Do what feels right to you. She has kneejerk reactions to you. Throw her offguard and keep her wondering.
Some days, ask how she is, other days, dont. Just be your wonderful self. She is an uptight, unhappy person. Dont let it get to you, if you can help it, ya know.