I'm having a really difficult time isolating the drop cap option to only one paragraph. Whenever I apply this effect, it changes every character at the beginning of a new paragraph. I just want one paragraph to have the drop cap and the rest to remain normal.
Because you've redefined Basic Paragraph style - it's the "default" ParaStyle - you need to leave it alone and create new ParaStyle for your first paragraph - then even more styles and assign them to the rest of the text - as per formatting you want to achieve.
I'm sorry but those are basics how to use InDesign 😞 You need to read more tutorials 😞
Drop caps are part of a paragraph style. You need to define a style that uses the drop cap and one that doesn't... for instance Body Text, and then Body Text DC.
You can't apply a Paragraph Style-based aspect to only one paragraph and should not use such spot or override formatting in ID in any case.