Can I get Shadowmere and save Cicero?

There is no way to both destroy the Dark Brotherhood and obtain Shadowmere. If you choose to destroy the brotherhood (by killing Astrid in the shack) you won't be able to meet Cicero, and the quest progresses in such a way that you find the sanctuary and kill every single assassin present.

If you do choose to join the Brotherhood (by doing what Astrid asks of you in the shack), eventually Shadowmere will be given to you regardless of whether or not Cicero lives (if you spare him, you end up lying to Astrid). Every event from that point on will still happen (The Sanctuary gets destroyed), but the questline doens't end until you re-establish the Brotherhood in another location.