Ah Annoca.... twas the same the day we got the pup... as I said.. bittersweet. smiling, but with tears behind eyes & that horrible lump in chest!
It is totally natural, as the others have said.
its also a really really weird feeling. But its better than the desperate feeling... anything is better than that.
You'll be ok..... your emotions will settle down eventually, in the meantime your grieving will be balanced by smiles... it really does help
xx