For future generations, I am posting the steps I've taken

Editing the code to replace all instances of Panel with an instance of user
control will cause them to work correctly. The steps are
1. Set form's AutoScaleMode = dpi
2. Edit code to replace Panel with UserControl. (UserControl won't show up
in the toolbox, at least for me.)

Fixing Tabpages was a bit more complex.
1. Set form's AutoScaleMode = dpi
2. Set tabControl's Dock=Fill (could depend on your tabpage's location.
3. Change to VGA mode. The controls in the tabpages will not scale up,
giving you more room to work. 4. Create a Panel in the empty space of a
tabpage
5. Select all controls from the tabpage and place them into the Panel,
maintaining their relative position.
6. Set the Panel's Dock=Fill
7. Repeat steps 4-6 for all tabpages.
8. Edit the code, replacing all instances of Panel with UserControl.

Doing this for a brand new form may not be so cumbersome, but you should
remember to add a UserControl to each TabPage before placing anything on/in
it.

Nathan