Hi, and welcome!
I am in a very similar situation with my partner through soon 13 years. She fell deeply in love with a close friend of ours, and came out as poly just recently, asking for my permission to start a relationship with him. They are very much in love, with a very deep connection underneath. I went through (and by all means, am still going through) the fears and insecurities which arises. There's all kinds of stuff popping up, and fear is excellent at finding new ways to scare you. I have actually been at times amused at what my mind can create...
The only real tip I can give you, is to sit down and have a good long think about yourself and your relationship, what do you want, what do you see as a long term goal. Then have a heart to heart with your partner and get the reassurance you need to feel safe. I had 2-3 weeks of constant heartache and churning guts over losing my partner. Now, only a month or two in, I feel completely safe with her. I know in my heart that she will not leave me for her new boyfriend. The one who can break this relationship is me. If your partner is clear and focused on you being as important in his life, and he has no intention of leaving you, you have an excellent start in this. If you also have a long-term goal of being with both of them, then it's even better.
Face your fears, grab hold of them and drag them out into the light, use your love and your partner's reassurance to contradict and prove your fears wrong. *hugs*
Mono engaged to poly VanillaCrazyCake
Don't fear god, don't worry about death; What is good is easy to get, and what is terrible is easy to endure.