How Do I Go About Getting a Second MOS in the National Guard?

Challenge #1 - any unit that has 88M slots is very unlikely to have 35P slots. Without positions available, they will not have quotas for school seats, nor a budget to send you, nor any incentive to do so. Your first objective needs to be to find a unit that has such positions, and is willing to train you to fill that position. This is not a short school, so it is a sizeable investment for them. Even units with a need may say no. As noted by others, you might need to look at the Army Reserve to find such an opportunity. Get with a career counselor to determine where (and if) such positions exist in your area.

I would suggest a different path. I would first attempt to pick up the 09L (translator) MOS. This should be an easier transition. At the worst case, they would want you to go to the six week school. However, I wouldn't be surprised if an OJT/Prior experience waiver were granted given your background. I am not a HR type, but it would also seem to me the sort of thing you could get added as a secondary MOS without HAVING to move units (they unit may not have a slot, but it would definitely be a good ability to have within the unit, if the cost were not excessive. At least that's how I would see it if a Soldier proposed it to me.)

Once you have the 09L, you will be more likely to be seen by "the system" as a language guy, and compete for assignments in that field, including eventually reclassing to 35P, if you still want to.

You need to look at the Reserve as well. Guard doesn't have much opportunity for language proficiency.