I know that Shadowmere is a quest reward from the Dark Brotherhood questline. However, since I took the other route, killed Astrid when she visited me and got the quest to destroy the Brotherhood, which I later did (she can't tell me what to do!), I was wondering if it's still possible to get her horse some other way and not through the quest.
The simple answer to your question, without cheats, is no.
The game is specifically scripted to give you Shadowmere once you have completed the Dark Brotherhoods quest related to the horse.
With cheats, you can get Shadowmere however. First open up your console, and type:
It should come up with an "ID" for the NPC for Shadowmere, this is what you want to spawn in.
Then, type in:
player.placeatme 0009CCD <- Replace with the ID for Shadowmere, although I'm sure that is the ID.
Shadowmere will spawn in next to you, now select Shadowmere by clicking on him/her/it while the console menu is open. Now type in:
And Shadowmere should be yours.
I understand that you want to play the game legitimately, and I'm like that too. But if you really want Shadowmere, perhaps a good way to "roleplay" it would be to say to yourself that you found him/her/it while destroying the hideout and tamed him/her/it? Up to you.