I'd sure vote for getting a room. You're down by 10, no checked bags, you can be in your room by a little after 11. Print your BPs , if you're not doing mobile, showered and in bed by midnight. With Precheck and no bag drop, I'm comfortable getting to security a hour before departure for domestic, so for me the alarm buzzes at 6. Hot shower and 6 hours of sleep vs bunking in the airport.
There's an Embassy Suite, Hampton Inn, Courtyard all about 10 minutes away. They're all about what you'd expect from those brands.
I'd book through the front desk noting the late check in and shuttle pickup. And check if it's a shared shuttle and stay at the starting point.