I only have a single drum kit, the SoCal Kit, in my Drum Kit Designer in the trial of Logic. How do I get other drum kits?

I stand corrected re trial limitations then. Live and learn eh

That being said, I don't think "Download All Available Sounds" is a viable advice in this case. I'd still insist on:

Logic Pro > Sound Library > Open Sound Library Manager > open "Drum Kit" and tick "Stereo Drum Kits" and / or "Multi-Output Drum Kits" to download the rest of DKD sounds.

It's quite normal for a trial version of software to be limited in some ways and that may include Logic's sound library. However, if you only installed the trial with just essential sounds, what you experience is not out of the ordinary.

Please, check: Logic Pro X > Sound Library > Open Sound Library Manager. This should open a window where you'd see which loop packs and other sound content are installed and which are available for download. But again, yours being a trial version, there might be some limitations.

There are no limitations on Apple trails.

All Apple pro apps have extra content that must be installed after the app itself is installed.

Go to the "Logic Pro" menu, to "Sound Library" and chose "Download All Available Sounds". Then go do something for the rest of the day, it will take a long time.