If you have a problem deciding if this is a sin, it would seem that you might have a problem deciding if any specific act is a sin.
So, to help out, it may be best to define what is sin.
Sin is any act which separates us from God. Not any act which makes us feel spearated from God, but which actually separates us.
The 10 Commandments are a good place to start; they do well to define general areas of sin (and a good examination of conscience which is based on them is helpful); add to that the requirements by the Church, and for most people, you have pretty well encompassed them.
God gave us the world, and we are expected to treat the world with respect. That includes pets; they are not meant to be harmed intnentionally, nor to be harmed carelessly. So if you have a pet, it needs clean water, food (both at appropriate times) and shelter. If you don’t have time for the pet, or cannot provide at appropriate intervals for its needs, then out of respect, it should be given to someone who can. While there is always the possiblity that the animal will not find another owner and be euthanized, if your purpose is to try to get it to another owner, the Humane Society is a good way of doing so.
What might be wrong is obtaining another pet before you can rectify whatever it is that prevented you from keeping this pet, as obtaining a pet when you are not able to care for it is not showing the pet (and ultimately, its Creator) the respect it is due.
And no, I do not support PETA.
But the bottom line seems less about the pet than about your understanding of sin… Hope this helps.