That's fine! He's allowed to doze off. She'll have some time to think, while he naps.
Clearly she's going to need to pick up some hobbies while living with a dragon. She reviews the ones she already has for whether they'd be much good while she's living with a dragon. At the very least, she should probably ask him to make her a harp so she can polish her slightly languished musical skills. Embroidery had been a bit too sedentary and time-consuming for her tastes, even if she'd liked the finesse and detail, she'd had too many other things to do. Maybe it's more viable if she combines it with snuggling. Since she's already staying still so she can cuddle a dragon.
Most social hobbies are out, but not all of them. There are probably a few card games she could teach him, so they can play cards together. There are definitely some two person board games, though maybe they have different tastes there. Hmm, what other social things does she have available? She supposes she could be ridiculous and suggest braiding each other's hair...
...
Actually now that she thought of that, she kind of wants to. She spends a minute considering the wisdom of braiding a dragon's hair while he sleeps. Soon enough, she decides that she's his favorite person partially because she's kind of silly sometimes. It's not like she's going to go as far as to make him look absolutely ridiculous by putting bows in it. Mostly because she doesn't have any ribbon, but still. Braids can be perfectly serious. Plus, while cuddles are extremely comfy, they're not very mentally stimulating. Neither is braiding someone's hair, really, but that's at least stimulating in a more tactile way.
Gently, she combs her fingers through his hair and starts carefully making little tidy braids. There's nothing to tie them with, so they'll fray apart, but she doesn't really mind.