So, I tried adjusting Negative LOD Bias in my Catalyst settings, but couldn't find anything like it. I also checked a program called RadeonPro that allows us to do it manually, but FFXIV shows a black screen of death when the program is active. I think it's safe to say Square intentionally maximized draw distance, probably to make the game look as visually appealing as possible.

Looks like I'll be lagging until I can afford an upgrade. (Or playing another game.)