Hiring/Onboarding Question

I’m a PT with 11 years experience (4 in home health). I interviewed with a new HH company eight days ago. The Director of Operations immediately said she wants to hire me. We’ve had email exchange daily for the past week about coverage area, and I’ve agreed several times. She’s said several times she’s going to get me an offer letter (two days ago she said she’d reach out to HR and have the offer letter in 24 hours; yesterday she said the same; today I’ve heard nothing). My question is whether I should reach out to HR directly and ask whether it’s best for me to direct future communication to someone in HR so I’m not bothering the local operations director? My wife says that’s too pushy/desperate. I say every other company I’ve ever worked for has either promptly told me “no” to hiring me or they have promptly sent me an offer letter (within two or three days). What do other people think about how to proceed?