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Don't ask her out in an official capacity. Make it seem really casual. Like tell her you are bored and hungry and see if she wants to go get some icecream with you.
Aside from that, don't worry about taking her out anywhere. Just escalate whenever and wherever you are with her--whether it's studying or playing sports with her. Start teasing her or saying things in which she'll have to respond in a certain way, then push her, grab her, or hit her when she responds. If done playfully she'll be forced to laugh. Now you guys are having a good time together laughing and being physical. Then the next time you do something physical, it won't seem awkward or forced...especially when it's a little more physical than the last time. Eventually you start bear hugging her, or putting your arm around her and pulling her into you. Basically treat her like she is your play thing. Don't worry, she'll either give you ioi's or iod's to let you know if she'll tolerate it or not. If she gives you iod's freeze her out. If she gives you ioi's, proceed.
Once you have established a relationship of physical contact with the girl, by the time you start going out on dates it'll be completely natural to hold her hand, hug her, put your arm around her, etc. You have to pave the road before you can travel down it. Playful kino is the groundwork for sexual kino. So I'd recommend going that route since this girl may be a tough nut to crack due to being shy and conservative.
This was really really good thank you! Could you just briefly touch upon the playful teasing and touching? But this sounds great and we'll have a week on campus without classes so it sounds like a perfect plan with all the down time.