So sorry for your loss. 
It will get easier I promise. I lost my beautiful 7 year old gelding this time last year. He was my pride and joy and it just about broke my heart. It was absolutely the right decision to put him to sleep but I still feel like I killed my friend. Like you there was relief immediately afterwards then the next few weeks and months were terrible. I can’t really explain how it gets better but it does. Little by little I realised I could go out to the field without crying, then I could look at photos etc. You know in your heart of hearts that you did the kindest thing for your friend and that, eventually, will be what carries you through.
This sounds ridiculous but I had this poem open on my phone for ages and it really did help, I can now read it without crying:
I'll lend you for a little while
My grandest foal, He said.
For you to love while she's alive
And morn for when she's dead.
It may be one or twenty years,
Or days or months , you see.
But, will you, till I take her back,
Take care of her for me?
She'll bring her charms to gladden you,
And should her stay be brief,
You'll have treasured memories
As solace for your grief.
I cannot promise she will stay,
Since all from earth return.
But, there are lessons taught on earth
I want this foal to learn.
I've looked the wide world over
In my search for teachers true.
And from the throngs that crowd life's lanes,
With trust, I have selected you.
Now will you give her your total love?
Nor think the labor vain,
Nor hate Me when I come
To take her back again?
I know you'll give her tenderness
And love will bloom each day.
And for the happiness you've known!
Forever grateful stay.
But should I come and call for her
Much sooner than you'd planned
You'll brave the bitter grief that comes
And someday you'll understand.
For though I'll call her home to Me
This promise to you I do make,
For all the love and care you gave
She'll wait for you, inside Heaven's Gate.