I agree. Each of us is a soul. Adam became a living soul when the breath of life (aka "his spirit") was breathed into him. Prior to that, God had created his body, but it was a body without life. I believe that a soul is made up of body and spirit. We have a body and we have a spirit; we are a living soul.
You said one thing in this post, though, that made me curious. You said we are souls and so are animals. But you don't believe animals will be resurrected, do you?
God formed man from the dust of the ground (Gen 2:7) and breathed into him the breath of life and thus the body of man became a living soul (Hebrew nephesh chay). However, God didn't stop there with humans.
Gen 1:27,
So God created man in his [own] image, in the image of God created he him; male and female created he them.
God is spirit (John 4:24). That is His image.
As originally created therefore, man was body, soul, and spirit. It was the spirit part that had fellowship with God. God, being spirit, can not be know by a body and soul individual. The creature of body and soul is limited to gathering information from their five sense which do not register spirit.
When Adam sinned he lost the spirit and thus his direct connection with God. God then used words, which our five senses can grasp, to tell us about the coming Saviour and how to get back to the original 3 part being. The whole point of the new birth is getting that spirit back again. Just before he ascended into the heavens, Jesus said we would get that spirit back again.
John 16:13,
Howbeit when he, the Spirit of truth, is come, he will guide you into all truth: for he shall not speak of himself; but whatsoever he shall hear, [that] shall he speak: and he will shew you things to come.
Jesus said the spirit would tell us the truth, not a body or soul.
1Thess 5:23,
And the very God of peace sanctify you wholly; and [I pray God] your whole spirit and soul and body be preserved blameless unto the coming of our Lord Jesus Christ.
God was speaking to born again believers so he could rightfully say body, soul, and spirit. A person not born again is called a "natural man" which Corinthians says can not communicate with God.
1Cor 2:14,
But the natural man receiveth not the things of the Spirit of God: for they are foolishness unto him: neither can he know [them], because they are spiritually discerned.
It takes a spirit to know a spirit.