I am attempting to change the path of my files, however the right side of the screen under Cache is cut off, so the browse button does not appear.

Yeah, that cut-off cache path window is a known annoyance in the latest Lightroom Classic.

Quick fix that works for most people:

• Press F12 (or go to Window > Screen Mode > Normal) to exit Full Screen mode if you're in it.
• Then simply resize the Preferences window by dragging the bottom-right corner bigger.

The Browse button should appear once the window is wide enough.

If that doesn't do it, try resetting preferences (hold Alt+Shift while launching Lightroom) – it usually brings the dialog back to normal size.

Let me know if the window still gets cut off after resizing.