Labor Condition Applications is specific to the employer and not to any future employee. Your new Univ.HR should fill this form out and post some location on campus for 2 weeks. If no objection from the public (usually nobody cares), they will proceed to your case to UCSIS.
Based on my knowledge, you should be able to work with the receipt (H1 to H1), but you need to get accurate answer from other experienced folks since you are from postdoc to assiant prof,a permanent position.