Can my family stay in our current house on post if I’m on unaccompanied PCS orders for a military school?

IF you know where you're going to go after your reclassification training, then you may be able to PCS your family to the new duty station. You'll go to your reclassification training and stay as a geographic bachelor.

The term that Housing was likely referring to is called "Housing Flexibility Authorization". However, that only applies up to 180 days after you PCS. It might be possible to get a two month extension on that (have to ask - could be told no). After you complete training, the government will move them from your current duty station to your follow-on assignment.

What normally happens in unaccompanied orders is that your family is authorized a housing allowance for wherever they are going to reside (their "designated place of residence") while you are on your unaccompanied tour. You stay in the barracks (or other government provided housing) and you get the normal BAH based on the zip code your family is at.

If you were staying off-post, it's a simple fix for DoD. Staying on-post (if space is tight at your location) becomes a matter of priorities. Since DoD has an option for your family (move somewhere, we give you BAH for that location, move to your spouse's PDS afterwards), you are now competing against the newly arriving family that is coming to your current installation that wants your on-post house - that's why it becomes a bit muddled.

There are a thousand "what ifs" that could apply, so I suggest you contact Military OneSource [login to see] ) and request to talk to one of the relocation counselors.

If your orders say unaccompanied, I would assume that Finance will treat the move like they would a TDY enroute. You should be given the option of leaving them at your current duty station, in which case you would get travel back to that location and then clear at that time, or you could move them to your follow on duty station like you would if you just PCS's, or you could move them to your TDY location at your own expense, which means you would clear your previous station. You need to make sure that your Command with still sponsor them and you will probably pull BAH from your home of record since you probably won't get government quarters for the family and the regs won't authorize non-availability.
https://www.benning.army.mil/garrison/dhr/Content/LEVY/TDY_OPTION_FORM.pdf

Look if orders say unaccompanied then they should be able to stay, ask the Garrison Cdr office- housing works for them.