You´re right, we never get to see any appointment! She arrives at the office before him, but "If I take you home, I´ll be late" indicates that he was going home before going to work (did I use the w-word?!). She could have had the time to get her hair done? Manicure (womancure)?
The most realistic thing would be that she just wanted him out of her hair.
I always believed it was just a line Maddie says to David. She freaked in the morning and he mentions making coffee so she probably figured that maybe David and her would lead right back to bed which she didn't want to do. She just tells him she has an appt. She can't even look him in the eye and tell him that. She needs to go behind closed doors to do that. We don't know how late they both were when they got to work .Maddie could have drove around and then showed up a little before David. Mind you David was almost always late and neither of them knew the other was there when they arrived only Maddie figured it out once she heard David singing. JMO