Lucy- welcome to this board. The things you mentioned seem very familiar to me , and I would guess also to other members on this board. That said, only a qualified professional can make this or any other psychiatric diagnosis. I would encourage you to have this same discussion with your psychiatrist (" last week you told me I had emotional instability...this has been on my mind all week...do you think that I have BPD? If so, what do you recommend for treatment?")
In terms of coping, or distress tolerence skills, there are many things you can do, whether or not you 'officially' have BPD. One source is a website called dbtselfhelp.com. You can read about the skills ,practice them, talk to your therapist about them - and see if they help you. There are many other books and websites as well.
Welcome, and good luck to you.