When the DX9 plugin was working there were graphical issues all over the place in Twilight Princess as with most games, far worse with the DX9 plugin than the OpenGL plugin.
I would suggest saving for a proper graphics card, anything Geforce 7xxx or equivalent should be sufficient. Another option would just be to play the game on an actual Wii, this game is a major headache in Dolphin especially with the massive slowdown in places like the forest and Hyrule field. I'm guessing you likely don't have a very powerful processor so the game would lag like hell for you and the slowdowns in Hyrule field and the forest would be more prominent for you.
I would suggest saving for a proper graphics card, anything Geforce 7xxx or equivalent should be sufficient. Another option would just be to play the game on an actual Wii, this game is a major headache in Dolphin especially with the massive slowdown in places like the forest and Hyrule field. I'm guessing you likely don't have a very powerful processor so the game would lag like hell for you and the slowdowns in Hyrule field and the forest would be more prominent for you.