Could it be, that you have in general problems with taking "disappointments" (Sorry, bad english.) at all. I do understand grieving for a pet, so whenever I lost one, i was absolutely sad. One of my cats that dies, was born in my house by an wildcat, I never had a cat, that got so in contact with us, as this one that lived with us since her first day. She sadly died out of an closed "digestive-tube". I am absolutely sad about it, and she was worth as much as any human friend to me. But to decide, that out of that experience, I should denie myself forever, to have a friend in my life again, because of them being able to die as well, simply does not make sense.
Enjoy the time you can enjoy, grieve when somethings end, but dont denie yourself to enjoy time again.
When it was about my issues about relationship, I did not let my grieve about it, disturb me having a nice time with other boys. There is nothing bad about it. I simply did not engage in relationships anymore, but that did not disturb being friends to them or spending time with others.
And the cat I mentioned will always be a very special one for me, because of her being so close to me. Still the cat that went here a year later, manages to make me happy. Its not about replacing someone, but finding an additional place in your heart. 