What do you usually say when you aren’t feeling well? Do you say, “I feel bad”?
If you write, “I feel badly,” you are in essence saying that your sense of feel is not working correctly. If you write, “I feel bad,” you may be saying that you are sick.
So you say “I feel bad”—not “I feel badly.” Wouldn’t you write, “I feel good” instead of “I feel well?” or even “I feel goodly?” Okay, so that’s a bit too much, yes?
I feel bad because I played the game so badly that my team lost.
This sentence is grammatically correct.
You could just circumvent the whole thing and say, “I am not feeling well.” When we write ‘well’ in this case, it means ‘not sick’ or ‘in good health.’