Can anyone provide advice on how to structure a proper request for reclassification in the National Guard?

As someone who's held 4 MOS's (11B, 21B, 37A, and 35D) plus 2 identifiers. The easiest way in the Guard and Reserves is to find out what units in your state or another have vacant 35F positions. If your unit won't work with you, the Guard and Reserve Recruiters should. You may have to travel out of state or be willing to transfer to the Reserves (The Reserve units in your state may have 35F's). Once you find out which units have vacancies (be it Guard or Reserves) then work with the Unit and/or the recruiter (if your transferring to the Reserves) to get an interview with the Unit/Units that have 35F vacancies. Once they accept you getting transferred is just a matter of getting your paperwork submitted and waiting on orders. I started in the Guard, Switch to the Reveres, Then Active Duty Guard (AGR one time casual orders renewed twice), then Active Duty (DC, Yemen, Germany (ECOM JOC)), and finally retiring from the Reserves after 34 years of service and enough point to garner an active duty retirement I have a bit of experience in this (I served 6 years Enlisted and 28 as an Officer)

I have actually gone through this before. If you want to re-class to 35F then look at my first answer. If you want to re-class to the 35M MOS, look at my second answer.

1) If you want to re-class to a new MOS that is not in your state, first step is to identify what states do have that MOS and have an active need to fill that MOS. You will then need to identify the NCOIC is that handles interstate transfers in the state that you would like to move to. From there it is just filling out packets and knowing how to speak with your chain of command. Once you have found the NCOIC, get a better understanding of how the process will work (As it can change from state to state). From there, go through your Chain of Command/NCO Support Channel (without going over people) and let them know what you would like to do, and how the process works (From personal experience, most Soldiers are unaware of how this process works)

2) If you would like stay in your state and just change your MOS, you have the best chance at doing this upon re-enlistment. When you re-enlist you have room to negotiate (through a retention NCO) on what your next contract will state. If you do not want to wait that long, then identify which unit in your state has that need. If possible speak to someone in that unit about how the process will work. For the most part, changing your MOS during your current contract is all about finding how what the request and approval process is and doing the paperwork. Ask how to transfer units, and speak to someone in that unit that would handle the process, and also see if they have space for a re-class. This is where problems normally occur, if you are not about to re-enlist, it may be a problem for another unit to take in a permanent party member vs. a new recruit. In a nutshell, you just have to request to change units, have a good reason, and that unit must have a re-class slot available (Meaning they have the funding to send you to a school).

If that unit has a re-class slot, then it is as simple as transferring units. If they do not have a re-class slot available, then you will either have to wait for your re-enlistment date, or put in an Officer Packet for the career field you want.

MY TIPS FOR YOU:
-Know how to speak with your Chain of Command
-Do not go over anyone in your chain of command
-Know the name and number of the Soldier(s) that handle each part of the process
-Be patient
-and REMEMBER: Most Soldiers (junior enlisted, NCO's, Officers, etc...) do not know everything. There are simply too many procedures for things like this, so make sure you speak with tact/respect to everyone in your chain of command and inform them of how it works. Generally this is a learning process for everyone involved.

Note: My response assumes you already worked with your chain of command when you said, "they kind of piled a bunch of NCO's in a room and pretty much, very respectfully told me it wasn't gonna happen and to get back to work." So the process I recommended is for working with an unsupportive chain. I'm also assuming they don't want you moving on for Unit Readiness and Unit Strength purposes. Which I totally understand but still wrong when dealing with soldiers and mentoring them career wise. If they were supportive your unit training Officer or NCO could help you find Units with 35F vacancies.