Your best shot might be for a January or February stay. The home resort window would be March thru July for a Feb stay. For example, for a stay starting Feb 21, you could book it with home resort priority until July 20, but anyone in
DVC can book it on July 21. Once the 7 month window opens, anything left in BLT will go quickly. So you may get lucky and get your points before the home resort window closes. This is your best chance. In 10 weeks, it will be mid June, so you will still have time for anything left at BLT.
If things get dragged out, you will be able to book any resort at the 7 month window. Get on line at 8AM Eastern 7 months out, and you will have several choices, but they will go quickly. A lot will disappear by 8:05.
Jan and February are not crazy months for DVC so your chances are good to get something you will like.
At BLT in Feb, there is still decent availability for a standard view 2 BR, although the last week in Feb is unavailable.
As for Oct, Nov, Dec - these are very heavy months, and your chances would be much less, but it never hurts to check. Saratoga Springs is a huge resort, so your chances would be best there, as you noted.