So I'm trying to figure out the best way to handle this situation with Recruiter who for months it looks like hasn't processed my paperwork?

So you go over, under, around or through him. With respect to this SFC, he is JUST an SFC in the works. He is not the be all end all. He has someone that he answers to. Go to that person. If you don't feel like dealing with that office, go to another recruiters office and present them with everything and repeat what you have said here. If that one is too lazy or incompetent or unmotivated, rinse and repeat. At some point you will find someone that will do their damned job and work to get you in.

As to your next step as far as this recruiter goes? You find out what the exact process is and document that. Then you straight up ask this recruiter where things stand and what the hold up is. You document his answer. If he lies or attempts to BS you, call him on it and take it to his command, to the IG etc. Just because you are not a never been touched before green recruit does not mean that you can be treated just any kind of way.

Can't you go to a different recruiter?

Purely as a thought, you could try the recruiting district office maybe, or ask chaplains, who I'm sure get far weirder requests for help, or your own chain, to try to intercede, maybe your three Congress local offices, their mil/vet staffers...obv, those are just thoughts purely winging it on the spur of the moment, certainly, you might also consider actually calling Recruiting Command, directly, just as another thought, as well, however, I'd very definitely consider asking for chaplain help, chaplains are handy often for weird problems like this, I've observed, honest....