Test Drive 2 works fine with hercules graphics for me. It sounds like you may be using an incomplete copy that only includes the EGA/VGA executable. There should be a DUEL.EXE launcher program and TD2CGA.EXE, TD2TDY.EXE, and TD2EGA.EXE executable files. TD2CGA.EXE supports hercules when you pass it a "herc" parameter on the command line, which is what the DUEL.EXE launcher program does.
OutRun also works fine with hercules graphics, but as with TD2 you may be trying to run the EGA/VGA executable because you have an incomplete copy. There should be an OUTRUN.EXE launcher program and OUTCGA.EXE, OUTTDY.EXE, and OUTEGA.EXE executable files. However, there is one version of OutRun that has broken hercules support due to an oversight in the launcher program where it doesn't pass the "herc" parameter to the launched executable. If you have an OUTRUN.EXE file that is larger than 1024 bytes and CHEVY.PES (CGA and Hercules), BEETLE.PES (Tandy), and CORV.PES (EGA/VGA) executable files, then you have the version with broken hercules support. As a workaround you can copy CHEVY.PES to CHEVY.EXE and run the game with the command "chevy herc".