I need some help. I created a droplet in photoshop to apply my basic photoshop edits. From Lightroom, I export into the droplet and it opens the files in photoshop to apply the changes. When I click "save" in Photoshop, the images are saved as jpg into the original folder, but my lightroom catalog doesn't update or see the new jpg file. It still shows the original DNG. Is there a way to save in photoshop after the droplet that will allow me to update the Lightroom catalog and access that JPG from there?
Precisely which Lightroom are you talking about? Paste the first lines from Help > System here.
With Cloudy Lightroom you can't do anything other than you are doing - at best you could send the files to PS, run the action, then save. But with Classic Lightroom you can set up the droplet as an external editor and files sent to the droplet will be catalogued automatically. As with many things, more advanced workflows are best done with Classic Lightroom.