You sound lovely, Lucy is very lucky!
I probably wouldn't lift up her house as it can be very scary for them and its a good idea to make their own little house a place where they can feel totally safe and secure- being adopted and meeting a new family can be very overwhelming!
When I adopted Gus Gus (a dwarf hamster) I took it very slowly and it took him about ten days/ two weeks to become used to me properly, but I started to notice his confidence growing within a few days. If you see her awake and wandering around, you could start by slowly placing your hand in her cage nearby her where she can see it and letting her smell you, then you can place a treat (sunflower seeds are usually popular!) on your hand and let her take it. That way, she feels calm and unthreatened in your presence, and starts to associate you with nice things
I wouldn't put a towel over her cage because, not only could she ingest it, it also she won't be able to see you and get used to all the sights and smells of her new home.
Good luck!
x