I'm so sorry to hear that this has happened. It sounds like you loved your pet so much.
I agree with Jess, your relief is linked to your anxiety and does not mean that you are heartless at all! My previous pet had seizures and a heart condition which meant that we had to put her down. Amongst the grief I also felt relief. Relief that my pet was no longer in pain but also that I didn't have to worry and be anxious all the time.
In my own personal experience when my pet passed, it meant that it was out of my control and for once I wasn't anxious about it. When the thing I feared the most happened, I was so used to being worried and anxious about it all the time that when it actually happened I was weirdly calm.
I used my own experience to try to not worry so much (easier said than done) but I really do believe our pets teach us life lessons that make us grow and learn.
I'm here for you!