I'm stressed about my host sister-- is it my fault or hers?!?

So I'm studying abroad in France; it's been 2 months and I'm here for 7 more. I get along with the rest of my family just fine-- I get irrationally anxious around them most of the time even though they're really open and friendly and funny, but that's something I can deal with and get over as time goes. However, the issue's with my host sister-- I'm not sure whether it's me or her, but we've been living together for 2 months and barely know each other. It's so awkward all the time. (continued)

Today we coincidentally caught the same bus and walked home together, and it was mostly silence and awkwardness besides a few questions I asked that only raised tension. I'm so frustrated-- I want to be close with her and I have no problem joking and getting close with other people, French or not. I make efforts to ask her out shopping or start conversation but I'm always rejected with polite coolness. When I first got here, she mentioned going shopping together. (continued)

I was a bit out of it b/c jetlag and homesickness but I definitely wasn't opposed to the idea; the plan never ended up happening and I'm not sure why. She just never mentioned it again. Since then, I've asked her to do things with me a couple of times/initiated conversations (asking how her day was, etc.) and her answers are always very polite and formal and one-wordy. We haven't ever hung out by choice or had a bonding conversation, and I've never met her friends. What am I doing wrong?!

I have a sister I'm really close with back home, so the fact that over here I don't have that kind of bond with a sisterly figure makes me want to cry-- I'm so close but so far away. What can I do!? Granted, some of it has to be my fault-- I get really anxious when I'm around her because I want her to like me so I want to act the right way (but I never seem to)-- but I don't do much wrong! My room's a bit messy but her's is more so, I don't use any of her stuff but shampoo, which I'm allowed to

I don't know what I'm doing wrong. I keep to myself what I should, I do my dishes and my laundry, I keep my room clean enough, I try my best to joke and act natural even though it's difficult. I'm too anxious and scared to confront my host mom-- what if she tells my sister and she hates me even more? I've never really talked with my host mom, or anyone in my host fam, really-- they never really ask about anything-- homesickness, comfort, issues, etc.

And it's all happening: I'm homesick, melancholy, frustrated with my nonexistent relationship with my host sister, and scared to talk to anyone in my host family in case it's not what they expect of me/they'd judge me. My host dad's really open, but I'm a bit scared of him, and my host mom is the nicest person ever but I also get the vibe that she wouldn't want to hear about the exchange student's issues. My 10yo host brother is out of the question, as is my h.sister. What should I do?

Sorry this is so long, I haven't been able to vent about anything to the people I should be able to vent to (host fam.) It's so stupid because I know it's my fault too, that me putting up a mask of cheeriness and "everything is ok" set a bar with my host fam I can't emotionally reach, and I'm scared to take of said mask for fear that they'll be disappointed with me or like me less for not being as optimistic as before. I want to be close to them, to share my life with them and let my walls down

But I just can't seem to let down my walls and in turn they can't seem to let down theirs. This long text matriculates to two main questions: 1. How do I act more natural and myself around my host family, and not be afraid to let my walls down and "let the feelings pour" (they literally don't know anything personal about me). 2. How do I become close with my host sister? I didn't expect to be best friends immediately, but I didn't expect two be total strangers after living 2 months. THANK YOU!!!