Okay, so I figured out the actual solution (for real this time), and will post in case this issue gets searched by another Sims 3 player with the same issue.
Apparently, EA has done the ridiculous thing of not releasing a patch that allows The Sims 3 to recognize modern day GPUs, so you have to do it manually by editing two files or use a third party mod.
With AMD cards, you have to go through the extra step of editing a line of code in a file labeled GraphicsRules, or else you'll have this clipping and shadows issue.
- Browse to Program Files/Electronic Arts/The Sims/Game/Bin/Graphics Rules
- Find this line: "setProp $ConfigGroup RenderShadowMode 1."
- Change so it reads " setProp $ConfigGroup RenderShadowMode 2."
That should fix the issue. Otherwise, you might have to update your drivers.