Yes. State constitutions can be amended to abolish all forms of slavery within the given state’s jurisdiction even prior to the passage of the Abolition Amendment at the federal level. That is, of course, exactly what happened in the case of human slavery, which had already been abolished in many states by the time the 13th Amendment was ratified.
To pass the Abolition Amendment at the state level, simply change the federal references to appropriate state references, such as:
Section 1. Neither slavery nor involuntary servitude of any animal shall exist within the state of New York the United States or any place subject to their jurisdiction.
Section 2. The state legislature Congress shall, within two years of ratification hereof, establish a system whereby the writ of habeas corpus shall be available to ensure the rights of any animal or group of animals…..
Passing the Abolition Amendment at the state level will help the abolition movement as a whole gain momentum. However, focus should ultimately remain on passing the Abolition Amendment at the federal level to avoid the slave state vs. free state dynamic and related problems.