Don't know if this helps you out bigdog, but I had problems fighting vasrai also with advent. I once saw a evacuator capital ship not only survive, but have their HP continually heal and rise during the course of one battle due to their Overseers. That was pretty incredible and frustrating to watch since I focus fired from about 60+ ships.
The problem here is that you were focus firing your entire fleet on a "tank" - a cap ship with full support is very difficult to take down because it has a lot of hull, gets full benefits from ~70% shield mitigation while being repaired, and your dps was probably suffering from the Stilaktus Subverters disabling your frigates and cruisers.
the way to attack a Vasari fleet late game, provided your fleet is at least of comparable strength (if they have 3x more ships than you do in terms of resource cost, your are most likely SOL anyway) is this:
1. Focus fire on the subverters ASAP. They will probably be "mingling" with your ships due to their "teleport and dsiable" ability, which should make them relatively easy to pick out. Take them out to the exclusion of all other ships - the group disable SEVERELY cripples your dps, but the subverters are fragile by Vasari standards and should go down in one or two salvos each if fire is concentrated - this also means they won't have time to gain benefits from support ships before being pulverized.
2. Once most of the subverters are taken out, you have a choice to make. If it's a mixed fleet, start trageting overseers, if it's an LRM-heavy fleet, start taking out the missile ships by focus firing a couple at a time. The LRMs are fragile enough that they won't have time to get healed if you focus fire them.
3. Take out the overseers if you haven't in step 2. If you have, take out the cap ship with the most annoying ability (probably the Carrier Cap Ship with repair cloud).
4. Mop up. If cap ships are a problem (i.e high level), take them out by focus fire. If the cap ships suck, take out the karrak assailants, heavy cruisers and karrastra skirmishers in that order (LRM -> heavy cruiser -> light frigate). Adapt this based on circumstances (if they only have a couple heavy cruisers mingling with your fleet, while the LRMs are sitting at range, pick off the heavy cruisers, than move on to LRMs, etc)
5. Kill the carrier cruisers last. Never waste your time shooting at carriers unless you are picking off remnants of a retreating fleet, or your opponent has nothing but carriers. They don't do any damage directly, have a large amount of shields and hull (meaning they get a lot of benefits from shield mitigation), and even when you kill them, the fighters/bombers will linger for another couple of minutes. Counter the fighters/bombers the best you can, and focus on ships that damage you directly.
Having a couple of Radiance Battleships (since we are talking Advent) can be good here - the battleship has a very synergistic set of abilities. Use Animosity to make the enemy attack your tank (Cap Ship), Ablative armor will passively restore your antimatter (while reducing DPS slightly due to the armor bonus), which you can use to detonate animatter in surrounding enemy ships, doing both decent DPS (albeit spread out, and so subject to repair cloud) and preventing the subverters and overseers from doing their thing. Cleansing brilliance is nice if your ship is that high, but really, the three primary abilities shine if used together.