Clean those pads *thoroughly* with isopropyl and solder wick. It'll smell horrible, like rotten fish, but it'll make everything go much more smoothly.
Also, continuity tests don't work like you think on capacitors - their purpose is basically to temporarily store a charge, and continuity tests work by applying a small charge and seeing if the other end picks it up.
Initially, a capacitor will fail continuity because it's storing that charge, and then it'll succeed as the charge is released. Checking for resistance is also not useful, as these have a low series resistance - almost like it's not there.
Your capacitors are probably just fine.