Salvaging a friendship from a mess, with limited time?

So, let me preface this by saying, I've always been, shall we say, "socially challenged". Never really was very good at making friends, never dated. Not particularly good at connecting with people.

In summer of 2012, I started hitting it off extraordinarily well with this girl I worked with (it's just a boring little retail job), and I fell for her super hard (for the sake of easiness, we'll call her "B"). I asked her out, and she said no. I was super bummed for a very long time because I liked her so much, and other coworkers picked up on it and got involved, started gossiping, and making a big deal about it. She understandably did not appreciate this, and I even got in some minor trouble at work because of how overblown the situation had gotten.

She came to me to have a discussion about it (this discussion took place about January of 2013), and I took responsibility for all the bad stuff and apologized to her. I didn't know this then, but at the time she had this discussion with me, she was seeing someone else. Being that I didn't know that, I told her that I wanted to be friends with her, I wanted to be able to actually be friends with her outside of work. She basically shot down that idea, and when the conversation was over, she asked me to never talk about any of this stuff again.

We were very distant with each other for the next several months. Then, in summer of 2013, she had a messy breakup with her boyfriend after he cheated on her. She started becoming a bit nicer to me, though we never really became as close as we were before I asked her out. She did become really good friends with this other guy we work with (a guy she actually dated before I even liked her; we'll call him "T"), to the point where she tells him absolutely everything about everything. I've been enjoying getting to be a little more friendly with her than I was for the first half of 2013.

Now, though, she's on her way out to a new job. She started a couple months ago, though she still works one or two shifts a week at our store until we hire someone new to fill her position. When I found out she was leaving, I was very saddened, because there really isn't anyone I enjoy talking to and spending time with as much as her. I wished that we could be friends, so that even when she's on to a new job, we could still talk and hang out. T suggested I talk to her about staying in touch, though I was nervous about doing so.

On the day I thought she was leaving, I asked her if we could keep in touch when she's gone, and that's when she informed me that she'd still be working an occasional shift with us; then she added "But yeah, sure", and that conversation ended. A few days later, T had told me that she told him about how I asked her that, and that she said she thought it was "really cute" that I did. On one hand, that made me happy, but on the other, I've always heard that "cute" is generally bad (as in, condescending, patronizing, etc.), coming from a woman, so I wasn't really sure how to feel.

Since then, though, neither B or I have brought up the whole "keeping in touch" thing, nor have we established any kind of friendship beyond work. Due to her limited schedule with us, I barely see her at all (I haven't seen her for three weeks, at the moment), and it's only a matter of time before we find someone to take her position.

So, I guess I just wonder what to do with all of this. I'd really like to have some kind of a friendship with her, but I don't really know how (again, I've always been pretty "socially challenged"). Not to mention, given the past, I feel like I'm walking on egg shells around the subject, because I don't know what she's open to. Back then, she didn't want to be friends with me and asked to "never talk about that stuff again", yet recently, she said yes to me about staying in touch and told someone else that she thought it was cute that I had asked her. It's just kind of confusing, and I don't really know the best way to approach it. I don't want to push this the wrong way and upset her (and ruin any chance of us being friends), but at the same time, there's not a whole lot of time or opportunity left for she and I to talk about this and figure it out.