I'm sorry that she didn't reply, I would guess that probably right now she doesn't see any good solution as your schedules do not match but she still should have replied... I think that sending another e-mail when you already know the plan for June is a very good idea and that's what I'd do. For sure you are neither a burden nor annoying, you met only once and you are not "bothering" her with all your problems but just try to get the scheduling part right - that's a very positive behavior not negative in my opinion... But I understand the feeling, I also felt similarly when I had some problems with finding a good time slot and in one of the e-mails I apologized my T for bothering her, but she responded that it is not bothering at all as comparing the schedules is very important...