That is very sad. It's so, so hard to lose an animal that has been in our lives.
I think that one of things I could never do in active alcoholism was 'sit with' my feelings. I always had to DO something to avoid them. Drinking or other acting out behaviour to distract myself from the real issue, rather than dealing with it.
I suppose grieving is pretty much sitting with our sadness. Accepting that feeling as natural and something that is necessary so that we can mourn a passing (Human or animal) and our personal loss.
One of my teenage pupils (SEN) whose mother died, is very good with other pupils (and staff) who are going through loss and grieving and I've listened to his advice to others a number of time. I wish I could send him to you. As I can't I'll pass on the jist of what he tells the other pupils -
"It hurts like hell, but you can't run away from it cause it'll catch up with you and be even worse."
"Talking to people helps. I'm not keen on hugging and all that stuff but if you don't mind it and you've got someone there for ya then try it and see if it helps."
"Crying is fine. But it gives you a sore throat after a bit so drink lots of water. "
"Remember them. You can write letters or make a picture with photos if you like art."
"We don't forget. But it stops hurting so much to remember most of the time. There could be bad days out of the blue though. That's fine. It's just natural."
Wishing you condolence, and strength in your recovery throughout your sadness. BB