After turning in all of the required marriage paperwork to my S1, 4 months ago, what is the next action that needs to be taken?

You need to do a 5960 if you have not already

From the finance standpoint it doesn't matter if your wife is enrolled in deers or not. What finance will need is the marriage certificate and 5960. 4 months is a long time. Either your S1 isn't doing their job or finance isn't. I recommend getting with your first line supervisor or 1SG and have S1 track it down. If they say they've turned it into finance then check with finance. If finance says they dont have it then go back to the S1 and ask them to give you a copy of the Transmittal Letter (TL), email, or whatever means they used to turn it in. Bottom line - there should be a paper trail showing that.

Nothing can happen until DEERS is updated. The spousal relationship must be correctly recorded there. Then spouse is officially a dependent. A dependent ID card can be issued and finance can do its thing. IMO you should: check DEERS, check MyPay. Update DEERS as much as you can. Same with MyPay. If the S1 needs to update DEERS be sure to check with them frequently. If you can’t update MyPay then go talk with finance. If all that fails, go see your first sergeant or CSM.