Asking a shy girl out

I've been getting to know this girl for a couple of months. She's not the most outgoing person, she's pretty shy and I think a bit green when it comes to relationships. We've hung out a few times, shared laughs (she's definitely opened up a bit because we tease each other and stuff), and there's been some touching, like grabbing my hand to check out my watch, hugs, and a time I touched her knee while talking she didn't back off.

Here's the thing: she seems genuinely happy to see me when we're together, but then she disappears for a few days. I've tried dropping hints about my interest, like suggesting she shows me around her hometown (she goes every weekend) or maybe inviting me over. The thing is, every time I bring it up, she kind of freezes up (Unfortunately, I can't bring her to my place right now)

Any thoughts on what to do in this situation? I want to make it crystal clear that I'm interested in her as more than a friend. We've already had lunch together, so the usual "let's grab lunch" might not cut it because she might miss the signals. I'm considering having a conversation with her about how I feel because honestly I don't know how to proceed. I'm on the introverted side myself, and the girls I've been with were usually very straightforward and active in showing their interest