How are school dates determined for an Army Reservist reclassifying to 68C?

SPC (Join to see), in the reserves, each major command gets "quotas" based on a number of factors for PME and MOS generating schools. Those quotas represent seats in each course that are disseminated down to individual Brigades (some are attributed to company, depending on command policy and if it is SQI/ASI generating).

Units try to get people to commit to each seat as soon as they get the allocations for the next TY and get Request for Orders (RFOs) submitted to lock in funding in October. If there are no RFOs for a seat 60 days out, BDEs lose the seat and the command gives it to someone else.

What will happen is your ATRRS operator (Training NCO) will put in a seat reservation request for you and you will end up in a "wait" status. Make sure your Pre-Execution Checklist is accurate and complete and signed! (including DTS authorization approved and RFO submitted), but you won't likely hear anything until October and it may not switch to "reserved" November/ December.
If you don't have a FTUS at your Det, you are being serviced by the Battalion, feel free to reach out to their Training NCO. If you are dead set on going, ask them when you need to submit the Pre-ex, DTS authorization, and RFO in order to get them to get the seat flipped to reserved status.

SPC (Join to see) it depends on the availability at the schoolhouse and the slots allocated to your command. Each org has a certain number of slots for schools each FY. Since we are nearing the end of the FY, there might be a little bit of a delay while the next year's schedule is finalized and allotments are made available.

CSM Everroad laid it all out.