I've installed it, loaded previous save games, none of them have a ghostly carriage in Dirtwater.
Started a brand new game, zipped all the way through it... at no point is there ever a ghostly carriage in Dirtwater.
Installed the DLC file all over again, just to try it... no ghostly carriage in Dirtwater ever, anywhere.
Extra details could be appreciated, especially if you just updated from Galaxy, tried installing on release day and so on. Is it the Windows, Mac, Linux version?
At least here's what I can say about the Windows version :
I had the same problem until about half an hour ago, when I had a completely stupid idea (you know, the kind so basic it worked).
I located my saves, copied the save folder (ie c:\users\<name>\AppData\LocalLow\Asymmetric Software\West of Loathing
-only the bold part) away on the desktop, removed the old backup saves, removed everything except my active saves, their backups and the "permaflags" file.
I then made a full uninstall of the game (ie click "no" when the unistaller asks if you want to keep you saves, so that everything is deleted and cleaned).
Then I downloaded the installers once again (with a cleared cache on your browser maybe it'll help, before I had 2 different 32bit installers and one 64bit installer, now I have a proper one fo each Windows version).
Put the saves back into the proper AppData folder.
Ran it. Exited after getting the title screen and checking my latest saves were available.
Then I installed the DLC and exited the installer did no click Launch, although I doubt it'd make a big difference).
Finally I ran the game again, picked my main save (almost beaten the game), went to Dirtwater, and all was well and good. The ghost carriage was parked right where it should be, close to the Saloon and the General Store.
I'm currently exploring the hedge maze and got through some great jokes and powerful items. Already thinking of getting them silver-plated, just for kicks.