Hi there~ Former nanny here....
I was also a "live out" nanny who would go to the family's house each day. I was there 40+ hours a week, depending on if the mom was traveling or not. That intial "getting used to each other" period can be tricky. I'm leaning towards it being a situation where she was embarassed and lied to cover up using the washer and dryer for her own stuff. I used to do my laundry at the family's house too. I was 20 when I started with them, and while I still lived at home, my mom and stepdad didn't do my laundry, I was expected to do my own and they got mad when I left stuff in the washer or dryer. Being at work so much, and having the active social life that a lot of young people have, I was hardly home to do my laundry, so it was super convenient to do my wash at their house during down time. I usually had a laundry basket in the trunk of my car, and I brought my own detergent and fabric softener.
When I first started with the family I worked for I was soooooo nervous about eating their food, answering their phone, watching their movies (one of their kids napped for 3 hours a day). I think you and your nanny are in the getting to know what's what phase. I worked for this family for 3 years and by the time that I finished working for them (I was going back to college fulltime) we were at the point where I house sat for them when they went away, they gave me a gas card to use to fill my tank since I drove the kids around so much to activities, I would pick them up/drop them off at the airport etc. They came to my wedding when I got married. I still keep in contact with them (the oldest is now in college!). It was a great relationship all around.
SO, I, personally, would give her the benefit of the doubt. Maybe come home early one day when she doesn't expect you, or have a family member or friend stop by to drop something off if you are concerned about what's going on in your home when you aren't there. It is very hard to find people you trust to care for your kids, as a parent now I see that. I really loved the kids I had and I really loved working for them, and it sounds like your nanny likes you guys too.