For me the keepers are Giroux, Backstrom, Tavares, Kovalchuk, E. Staal, and Rask. The first 3 are gimmes, and I think Kovalchuk is going to be fine without Parise. He was great for years in Atlanta without much help, and he's been tearing up the KHL this season. The changes that Carolina made can only be good for Staal, and Rask is being hailed as the second coming.
If you're really sold on not keeping any goalies then why don't you try to move them? Team #5 in particular looks like it could use some help in net. Even with a weakened system in Montreal I'd much rather have Price than Brodeur, and in 2 years Montreal could be back in contention whereas Brodeur will have retired. You might even be able to pull it off by offering Kiprusoff, but even better would be if you could move Rask. Surely there would be a market for a guy like that. If you could swing something like Price for Brodeur and a decent pick, or even Price and a pick for Brodeur and a pick upgrade, I'd say it's worth it. You could target team #7 and Luongo with the same sort of thinking, but I'd expect it to be much less successful. In any case, it would be something I would explore.
The problem, for me, with moving/not keeping Price and/or Rask is that neither Iginla nor Getzlaf is an upgrade on them in this format. You mentioned being the only Flames fan in the league which to me means that you'd be able to redraft Iginla without much issue, because the other managers will see a 35 year old, 67 point player. Getzlaf had a terrible year which means he will fall farther than he should. I would be pretty surprised to see either of them go before Price or Rask if they were all available in the draft.
By all accounts you've got the best 6 keepers, top to bottom, in the whole league regardless of who you end up keeping.
You mentioned that you came in 2nd last year. Who came in 1st? Was it team #6?