It is very important that you and the counsellor “gel” but sometimes you just don’t. I went for counselling a few years ago and the woman was also not very good at timekeeping. I got there one day and waited and waited and another man was waiting. She came out and said “whoops, it looks like I have double booked.” She then said she felt the other man’s needs were more urgent than mine so she would see him and I had to make another appointment. This was NHS counselling.
I think I would just ring the counselling dept of this charity and be honest and ask to see someone else and say that as you have already had appointments cancelled you would like it to be dealt with urgently.