Sounds like you may have done this already with your Monday coffee date, but maybe you could try just texting her out of the blue and say "Hey, do you want to go see a movie this weekend?"
It's a classic avoidance technique for people to say they're busy in order to avoid meeting with someone they don't want to see. By no means am I suggesting this is what you're actually doing, but it's possible she could be taking it that way. By initiating a request to make plans with a specific date in mind puts the ball in her court and makes it clear that you really do want to hang out.
Something else I just thought of is you could try some little token gestures, like buy her a little something that you think she'd like, and say "I saw this in the store and it made me think of you, I just had to get it." Or make her a mix-tape of some songs she'd like. That sends an UBER clear message that you accept her and want to be friends.
From the caution you say she's shown, she's obviously concerned about harming your relationship or stepping on your toes, so anything you can do to make her feel accepted will help.
Gralson: my husband. Auto: my girlfriend.
Zoffee: Auto's husband. Cue: Zoffee's boyfriend. Bookie: Cue's wife.