An old and worthless dog must find a forever home before it’s too late.

I agree with Keymiser, what sets this story in motion and what are the stakes?

Not enough information, You need to establish in the logline what make us care about this worthless dog. What is at stake if the dog doesn't find him a home in time? the consequences of him or obstacles not getting his dream home? also, worthless is such a vague description of this animal....Is there an antagonist? and is that antagonist is physically something else, or mentally him finding himself and self worth, because if he's worthless he has to find a way to NOT feel worthless, give us something that show he willing to change and question HOW will he make that happen?

I would take a look at the old 80's Disney film "Oliver and Company" for some inspiration, seems like a similar theme here...just my two cents.