According to a side quest guide there are only 5 battle quests by chapter 9 which would mean you did 0 of them when you were given them. These will presumably be unavailable to complete post time skip. Some of your paralogues should have expiration dates after the the time skip (if you did not do these particular ones already). What I would do is make sure you are actually doing battle quests and not the optional battles that are just there for exp/gold/materials.
I was able to finish all of the battle side quests pretty easily doing 2 battle days per month. In the earlier part of the game you only need to do one exploration day since you do not have enough professor energy to burn through all of your students motivation for the month. I have only done a Golden Deer playthrough so I'm not sure if there are wildly different amounts of side quests in other routes.
To me this really just looks like mismanagement of free days but it can still be salvaged. If you take 1 free day a month to explore and ensure that all of your students have high motivation, you can finish all of your current quests. Make sure you check the expiration dates of your paralogue missions to see if they can be done post time skip.
In the future, if you do another playthrough or decide to restart your current one, try to follow a formula of 1-2 exploration days, 2-3 battle days, 0-1 seminar days (Only do a seminar day if you have ABSOLUTELY NOTHING to do on that day), 0 rest days as you shouldn't need the Sword of the Creator.
TLDR: Check paralogue battle expiration dates and only do one exploration day a month the rest battle days and you may just make it through all of your quests.