What have you been doing? I've yet to see any cute death screen. D: I've never tried blackmailing anyone, so there's a lot I might be missing out on by not doing that?
In early runs I went to Myrkka II first for the safe vote and easy quest. Then Myrkka I with enough Etiquette and Empathy to not make a bad impression, then Draug for the expert science advisor and his warp speed movement. I've taken a few variations here, and I've always ended up with votes to spare, though without much revealed of the deeper plot.
There are a lot of really good advisors in the Tharkas sector, so it might be possible to start in that direction and pick up a talented cabinet. Opposition from Myrkka hits fast if you don't leave good impressions early, though, and it's hard to link up with Admiral Yakan's patrol from this angle.
There's a lot that happens if Esmarai doesn't have the space to explain herself, is allowed to live, and you don't adopt the prince. It probably makes the game much harder, so you could avoid that route to make it easier. I think there's a lot of information that's only available this way though, but I'm still trying to thread that needle myself. >_>
I've made some pretty detailed diagrams and notes if need be, but I'm worried about just dropping them here without being able to hide them in spoiler text.