Let's see. I'm sorry if I missed something.
If sign A is true, then signs B and C must also be true. But this leads to a contradiction, since at least one sign must be lying. So, sign A is actually false. But if it's false, then I cannot believe what's written on signs B and C. Therefore, all three signs are lying.
Sign C says that I shouldn't take road B, but since it is lying, I choose road B.