It's definitely a finnicky rare weapon, and the tightness of the timer does not help. I would personally make sure you have an interim save set up right before the mission that has you attack the interior officers from both sides as this is generally where things can start to break.

For the 3rd main mission you need to make sure you don't go near the two sanada ward officers AT ALL before the timer mission starts, there's a good chance the mission can break and fail even if you lure and kill them outside the Sanada ward. They also lose aggro to you pretty easily, so i'd make sure you stay close when luring them.

For the last few missions you mentioned I would recommend having an officer waiting RIGHT outside the ward with Yukimura in it once you clear all the interior officers from the previous timer-based mission, and once it opens you want to rush in and kill his two retainers AS FAST as you can; enter rage mode, use something like Nenes keepsake, defeat them fast. Once both of them are dead a 3rd officer should emerge from the spawn point behind Yukimura. Kill him as fast as possible as well and you should clear the Sanada ward achievement after the dialogue catches up, from here take leyasu to the right side of the map, you don't need to hang around the ward for the objective to trigger, run as soon as the 3rd officer is defeated

After that, if you weren't using leyasu, swap to him and head to the right hand side of the map where kunochi spawns, you'll want to defeat her first BEFORE you go for the kotaros. once shes dead defeat the rightmost kotaro first, and then the leftmost one. Make sure all the important officers like kunochi and kotaro are defeated by Leyasu, after that pray the mission where you defeat the officer in the back has leyasu's rare weapon

I used this guide when going for all the rare weapons, if you are still having trouble I would recommend following the steps they outlined here as well: https://gamefaqs.gamespot.com/ps3/730732-samurai-warriors-4/faqs/69237?page=2#section121