DH, to me this doesn't sound like a comfort or rapport issue, but is probably related to attraction and control.

Firstly you have to build attraction by showing her that you have value (that you are fun and socially proof yourself) and that you aren't dependent on the outcome of your chat with her, whether she says yes or no. It sounds backwards but the less you give a shi*t, the more she will. Just don't mistake not being fussed with being arrogant.

Secondly take control, if you're getting IOIs don't ask her whether she wants a drink, ask her what she's doing on Monday/Tuesday evening? then when she says nothing say, "good, I know a great pub/cafe/bar/ice-rink that you would love, see you there at 8?"
People are more likely to be free at the start of the week and you're taking control and letting her know that you are the man