I'm sorry for your loss losing anyone is never easy. You didn't betray them. Euthanasia for an animal that is in pain or losing function is one of the most peaceful ways they can pass on. If you have any experience with astral travel that might be able to help, or if you have an alter for the deceased or for a deity that deals with passing on, or an alter for animals of some sort (I'm not sure what path you follow so it may vary) you could put some of their toys or favorite items on it. As for emotional support having a trusted person wether it's a family member or a councilor can be a wonderful and helpful thing for some people.
I am so sorry for your loss. I know how deeply you feel the loss, I have had to euthanize several of my cats and even though you can rationalize your actions, it still is heartbreaking.
Light a Brown Candle with your beloveds name carved on it. If it was a Cat ask Bast to take her into her care. If a Dog, I would ask Hecate to care for her. Tell her spirit you will never forget her and you will see her again in the Summerland.
Blessings and peace to both of you.
I found this little poem that may help you cope with the loss.
Request from Rainbow Bridge by Constance Jenkins.
Weep not for me thought I am gone
Into that gentle night.
Grieve if you will but not for long
Upon my soul's sweet flight.
I am at peace, my soul's at rest
There is no need for tears.
For with your love I was so blessed
For all those many years.
There is no pain, I suffer not,
The fear now all is gone.
Put now these things our of you thoughts,
In your memory I live on.
Remember not my fight for breath
remember not the strife
Please do not dwell upon my death
But celebrate my life.
I know how it feels. but believe me, you didn't betray her, you took her pain away. My dog had to be put to sleep last year because he lost feeling in his entire body, except his head. all he could do was move his head. right before then he got terrible arthritis where it was already so painful for him to move. he was 9 years old. I felt like a part of me went with him when he was put to sleep. He died in my arms. but the first night without him, me and my parents kept getting strong signs his spirit was here, we would hear barks, smell him, hear him walk up,etc.. and for a long time we constantly had dreams of him. I was in deep depression though because I wanted him back alive. we never wanted another dog, but 8 months later a family friend gave us a puppy. and on memorial day another dog (stray) came to us. after that we get signs of our deceased friend less often. I feel like till we had other dogs, he was here to protect us and show us he's ok. in life he was there when I did magick, and really anything else. he was huge (a pitbull mastiff mix) but was a sweetheart, and very protective. remember that your familiar is alive in a different way. you may not see her, but she can still be right there. whats important is the bond you two had, and you still have it now. I don't believe death causes bonds to be broken. remember that the loss is only temporary, because there will come a point in time where you two will reunite. you can pretty much say that we are all spirits that rent bodies so we can live on earth for awhile. she is happy now because she will never have pain again. remember that you are the one who has brought her this joy.
I am going through this exactly right now. Two days ago I helped end the suffering of my beloved Isis. She was my familiar. The only one I have had. Like your familiar, she led me to my path. She saved my life numerous times because my love for her and connection to her was greater than a need to harm myself during some very dark years. I feel I did not do enough and yet she stayed for 20 years and I adored her. She was always with me during my readings and rituals. I feel like I cannot breathe. I feel as if all of the magik I felt has been pulled from the world. I feel lost and the chaos and discord of the world is drowning me.
I just wanted you to know there is someone out here feeling what you are feeling. Maybe we can share our journeys back or at least find comfort in another Wicca feeling lost and without magic after the departure of her familiar.
Though, death is not the end and you can surely contact all of your familiars or even find them in the spirit rounds (either through astral projection or lucid dreaming). It is not the end.
I just lost my familiar as well. my condolences for your loss. Something that my wife said made me feel slightly better. They didn't die. They were given a promotion. They went from being familiars to spirit guides. And they are still and always be watching over us.