Should you admit guilt to a lawyer?

This is hypothetical I have not done anything. I have heard it is always best to diclose everything to your lawyer.

Lets say I am accused of hit and run. I have made no incriminating statements to the police, and have quickly repainted the car. The prosecution has a weak case (I could always “lie” and say some wierdo just took my tag down and made up hit and run so he could get the insurance to pay his damage, etc.). Now if I actually tell my lawyer that I did it, woulden’t they advise me to plea no contest or something as opposed to lie (I know lawyers aren’t allowed to advocate lieing)? If I just tell my lawyer that the wierdo took my tag down it will be up to him to prove I hit him and ran. With no cameras it is now my word against his and an easy case for my lawyer.

In the above case I feel lieing to the lawyer would be the BEST option. Do you agree?