Please help me with H1b transfer questions

Xiao baitu & ge wei da xia:
I am current an H1b holder. I want to apply for another job, but the job deion says that "the hired candidate must be legally authorized to work in the United States without an employer-sponsored petition for a visa, such as an H-1B visa. ".

Since I have my H1b, I only need the transfer instead of sponsorship for a new one. Do you think I am eligible for this job?

Besides, if the employer does not want to transfer the H1b, can I hire my own attorney for the transfer?

thanks a million