My recommendation is based on my own experience. I took the AWS Architect Associated first and then I took the Professional Level. Do you need one to pass the other? the answer is not, does the associated level helps to pass the Professional level oh yes!
My advice is for you to consider taking the associated first and then the professional. In relation to the specialty, I can't say much since I have not sat any specialty AWS certification, all I have heard is that associate and professional levels are worlds apart from specialty which are very focus and narrow.