You're not alone! I'm sure we've all been there. I remember messing up some advising sessions in the past. The eye contact thing was always awkward. Imagine sitting there for 30 min - 1 hour directly facing the counselor and forcing yourself to look straight in her eyes the whole time. One counselor I went to actually avoided eye contact with me. I also avoided eye contact with her. When I looked down, she looked at me, but when I looked at her she looked down. It's like playing peek a boo. In another counseling session, I came in there very depressed because of the bullying I went through earlier - big mistake! The counselor asked me what's wrong, how was I, but I kept my head down the entire time and didn't tell her what's wrong. Then she got a bit angry because she thought I was lying to her about the test scores that I submitted, but she verified and realized I told the truth. The next time I came to her for registration for classes, she said I didn't have to come at all - just do it online. I also remember 1 particularly horrible advising session with this male advisor - he just completely blew up in front of my face. I think it was something I said that pissed him off - he thought I was lying so he got angry and started yelling at me. He said something like I couldn't fool anyone with my cute face. By the time he was done yelling, I was on the verge of tears, my face buck red. He then calmed down and stopped talking. When I got out of there, I went to the bathroom and cried. I never went there for advising again.
So those are just some personal experiences. If you're not comfortable with your current counselor, I encourage switching to a new one.