The game may still be configured to use the ID for your previous card (something about "GI not found" or "invalid argument in the init string" may show up in the windows event log when the game crashes).
This is a common issue when a system has two GPU's also (the CPU's onboard graphics and a drop-in PCI-E card). Make sure that the standalone GPU is set to handle all 3D graphics. The GPU's control panel may have a profile for the specific game still if you had created one with your last card and are still using the same name brand as before. If so, make sure to update that profile to point to the new card.
You can try resetting the FFXIV config as well. At the launcher where you sign in with user/pass/token, click the gold Config icon along the bottom. There is an option there to reset the system settings. First, use the backup feature to save your character settings to a location on your hard drive (just in case... never underestimate the value of a backup). Then use the option to reset the system settings. This shouldn't affect your character settings, but you never know (why you made the backup). Ideally, this should reset your display, controller, and sound options as if it was a new install but preserve your character specific data (if character data is reset, load the settings from the backup--that file is only supposed to be character profile data if I remember right). This should allow you to set the proper GPU in the display settings now if that is where the problem was.
Lastly, there is a chance it could be a problem with the DX9c files. There was an old glitch with the web installer where some key files would not get installed on Windows Vista or newer OS. IDK if Microsoft ever resolved the problem. The old workaround was to use the offline installation package. If you go back to the Microsoft page for the DX9 installer, there should be a section further down the page for downloading the end-user runtimes package for offline/standalone install. It is about a 95MB package, and the installers in that package will unpack and copy/register all files in the runtimes (unless Windows File Protection catches a newer version of the file already present/registered). It's 4 or 5 executables that you will need to run (unless they've streamlined it since I last used it).