Switching to new employer - PERM, I140 process

Looking for advice on a situation I'm dealing with.
I’m in my 7th year on H1B, my I-140 is approved, and my priority date is locked.
I’m planning to switch jobs, and my new employer is willing to transfer my H1B. The catch is, they want me to handle the GC process. From what I’ve researched and the legal advice I’ve gotten, it seems like the employer is supposed to take care of the GC process, especially the PERM stage, because it involves recruitment, and the employee can’t pay those fees.
However, the employer isn’t obligated to pay for the I-140 or I-485 stages, and those costs can be handled by the employee. Has anyone been in a similar situation?

Would love to hear about your experience!