It's natural to be affected by the first session in the way you describe; this is all new to you, and no doubt you are touching on difficult stuff so it's pretty scary! My advice would be to take your time and pace yourself. You don't have to disclose everything at once. I've been seeing my T for 16 months and I'm only just starting to touch on certain topics.
Do you mean you have to leave this therapist in 2 months? If so I would consider waiting till you're settled somewhere before you look into therapy long term. It would be hard to just get going at the point where you have to stop because you are leaving the area.
If you are continuing therapy right now, it would be worth talking to your therapist about the strong feelings which are emerging.