Once I download the fonts into Design Space, they don't look like the sample. They are usually missing all of the curls and swirls. Should I be installing the otf or the ttf font?

The curls and swirls are most likely alternates. It does not matter whether you use the TTF or OTF. What format to use is your preference and which works best with your program. Your program must support OT features, or an alternative method like a glyph palette, to access those alternates. Microsoft Word has insert symbols to access those alternates. See if your program has a similar feature.

Thank you. for your reply.